0$ and $m\\in (0,1)$,\n\t\\begin{align}\n\t\t\\label{eq:1-moment}\n\t\t\\frac{(1-m)e^t}{e^t -e^{mt}} \\int_0^\\infty u\\: \\mu_t({\\rm d} u) = 1,\n\t\\end{align}\n\tand\n\t\\begin{align}\n\t\t\\label{ineq:2-moment}\n\t\t\\frac{(1-m)^2 e^t}{e^t-e^{mt}} \\int_0^\\infty u^2 \\: \\mu_t({\\rm d} u) \\leq \\frac2{1+m}.\n\t\\end{align}\n\\end{corollary}\n\\begin{proof}\n\tThe equality \\eqref{eq:1-moment} directly follows from Corollary~\\ref{cor:m-3} with $k=1$. To prove \\eqref{ineq:2-moment},\n\twe observe that\n\t\\[\n\t\t\\int_{mt}^t \\int_{m u_1}^{u_1} e^{u_0} {\\: \\rm d} u_0 \\: {\\rm d} u_1 \\leq e^t-e^{mt},\n\t\\]\n\tthus the inequality is a consequence of Corollary~\\ref{cor:m-3} with $k=2$.\n\\end{proof}\n\n\\begin{lemma}\n\t\\label{lem:mu_t_erg}\n\tThe family of probability measures $(\\mu_t : t > 0)$ converge to $\\mu$ in total variation distance, i.e.,\n\t\\[\n\t\t\\lim_{t \\to +\\infty}\n\t\t\\|\\mu_t-\\mu\\|_{TV}=\n\t\t\\lim_{t \\to +\\infty} \\sup_{B \\in \\mathcal{B}(\\RR)} \\left| \\mu_t(B)-\\mu(B) \\right|\n\t\\]\n\twhere $\\calB(\\RR)$ denotes $\\sigma$-field of Borel sets in $\\RR$.\n\\end{lemma}\n\\begin{proof}\n\tLet us observe that by Corollary \\ref{cor:m-3}, for each $t > 0$ and $k \\in \\NN_0$ we have\n\t\\[\n\t\tM_k(t) = \\int_0^\\infty u^k \\mu_t({\\rm d} u) \\leq M_k\n\t\\]\n\twhere $M_k$ is defined in \\eqref{eq:85}. Then by \\eqref{eq:84}, we obtain\n\t\\[\n\t\t\\sum_{k = 0}^\\infty M_k(t)^{-\\frac{1}{2k}} = \\infty.\n\t\\]\n\tConsequently, there is the unique measure on $(0, \\infty)$ with moments $(M_k(t) : k \\in \\NN_0)$.\n\n\tLet $Y_t \\equiv t$ for $t > 0$, and let $\\mathbf{X}^{\\rm TCP}$ be a process obtain from $\\mathbf{Y}$ by partial resetting\n\twith factor $c$. The probability distribution of $X_t$ equals $\\mu_t$ since they do have the same moments, see\n\t\\cite[Theorem 3]{MR2426601}. Therefore proving the lemma is equivalent to proving ergodicity of the process\n\t$\\mathbf{X}^{\\rm TCP}$. The latter is a consequence of \\cite[Theorem 1(3)]{MR1157423} together with\n\t\\cite[Theorem 2.1 and Remark B]{MR1956829}. More precisely, the process $\\mathbf{X}^{\\rm TCP}$ admits an embedded\n\trecursive chain $\\mathbf{Z}^{TCP} = (Z_n^{\\rm TCP} : n \\in \\NN_0)$ where\n\t\\[\n\t\tZ^{\\rm TCP}_n=X_{T_n}, \\qquad n \\in \\NN_0.\n\t\\]\n\tSince $\\mathbf{Z}^{TCP}$ satisfies\n\t\\[\n\t\tZ^{\\rm TCP}_{n+1}=c Z^{\\rm TCP}_n + (T_{n+1}-T_n), \\qquad n \\in \\NN_0,\n\t\\]\n\tits driver $(T_{n+1}-T_n : n \\in \\mathbb{N}_0)$ consists of i.i.d. exponential random variables, see\n\t\\cite[Definition 1]{MR1157423}. Hence, by \\cite[Theorem 1(3)]{MR1157423}, it is enough to show sc-convergence of\n\t$\\mathbf{Z}^{\\rm TCP}$, see \\cite[Def. 4, p. 23]{MR1157423} for the definition of sc-convergence.\n\tIn view of \\cite[Theorem 8]{MR1157423} it is enough to check conditions I--III given on page 18 of \\cite{MR1157423}\n\t(notice that the assumption $\\mathbb{E} \\tau_V(x)<\\infty$ is superfluous since it is a part of condition I). We notice\n\tthat the conditions I--III hold true if $\\mathbf{Z}^{\\rm TCP}$ converge in total variation to its stationary law,\n\tsee \\cite[Theorem 2]{MR1157423}. Now, let us observe that the Markov chain $\\mathbf{Z}^{\\rm TCP}$ is an autoregressive\n\tprocess of order $1$, and it is positive Harris recurrent, see \\cite[Theorem 2.1 and Remark B]{MR1956829}.\n\tSince $Z^{\\rm TCP}_n$ has an absolutely continuous distribution, see e.g. \\cite[Theorem 2.8]{MR1894253}, we immediately\n\tconclude that $\\mathbf{Z}^{\\rm TCP}$ convergence in total variation.\n\\end{proof}\n\n\\begin{remark}\n\tThere is a purely analytic proof of Lemma \\ref{lem:mu_t_erg}, however it would require preparatory steps which go\n\tbeyond the scope of this article. The details will appear elsewhere. For the sake of completeness, here we provided a\n\tprobabilistic arguments.\n\\end{remark}\n\n\\begin{remark}\n\tBy Theorem~\\ref{thm:weak_conv}, the measure $\\mu$ is uniquely characterized by its moments. Since it has the same\n\tmoments as the random variable $Z$ in \\cite[(5.15)]{Kemperman}, the density of $\\mu$ can be read of from\n\t\\cite[(5.9)]{Kemperman}, that is\n\t\\begin{equation}\n\t\t\\label{eq:81}\n\t\t\\mu(u)=\n\t\t\\frac{1}{(m,m)_\\infty}\\sum_{k=0}^\\infty (-1)^k \\frac{m^{\\frac{1}{2}k(k-1)}}{(m,m)_k}e^{-m^{-k}u},\n\t\t\\quad u \\geq 0.\n\t\\end{equation}\n\\end{remark}\n\nThe following lemma will be important in Section \\ref{sec:cyl}.\n\\begin{lemma}\n\t\\label{lem:inf_mu_t}\n\tFor all $t_0 > 0$ and $\\delta_2 > \\delta_1 > 0$ such that $\\delta_1 < t_0$ we have\n\t\\[\n\t\t\\inf_{t \\in [t_0,\\infty)} \\mu_t\\big([\\delta_1,\\delta_2]\\big) > 0.\n\t\\]\n\\end{lemma}\n\\begin{proof}\n\tSince the function $\\mu(u)$ is a density of the probability measure $\\mu({\\rm d} u)$ it is non-negative. By \\eqref{eq:81},\n\t$\\mu$ has holomorphic extension to $\\big\\{z \\in \\CC : \\Re z > 0\\big\\}$. Therefore, it has finitely many zeros in\n\t$(\\delta_1,\\delta_2)$. In particular, $\\mu([\\delta_1,\\delta_2]) > 0$. Hence, by Theorem~\\ref{thm:weak_conv} there is $T > 0$\n\tsuch that for all $t > T$, $\\mu_t([\\delta_1,\\delta_2]) > 0$. It remains to deal with $t \\in [t_0, T]$. Let $j \\geq 2$\n\tbe such that $m^j < \\delta_2 T^{-1}$. Using \\eqref{def:mu_t}, for all $t \\in [t_0, T]$ we get\n\t\\begin{align}\n\t\t\\nonumber\n\t\t\\mu_t\\big([\\delta_1,\\delta_2]\\big)\n\t\t&\\geq\n\t\te^{-t} t^j \\int_{(\\delta_1,\\delta_2)} P_j(u/t) \\frac{{\\rm d} u}{t} \\\\\n\t\t\\label{eq:82}\n\t\t&= e^{-t} t^j \\int_{(\\delta_1/t,\\delta_2/t)} P_j(u) \\: {\\rm d} u.\n\t\\end{align}\n\tLet us observe that \\eqref{eq:82} defines a continuous function on $[t_0, T]$. Moreover, by Proposition \\ref{prop:1}\n\t$P_j$ is positive on $[m_j, 1]$. Thus \\eqref{eq:82} has positive infimum on $[t_0, T]$. This completes the proof.\n\\end{proof}\n\nIn the remaining part of this section we prove auxiliary lemmas which are helpful in studying ergodicity of the processes\nwith resetting.\n\\begin{lemma}\n\t\\label{lem:unif_conv-1}\n\tFix $C, \\gamma > 0$ and let $\\mathcal{G}$ be a family of functions $g : (0,\\infty) \\to \\RR$, such that\n\t\\[\n\t\t0 \\leq g(u) \\leq C\\big(u+ u^{-1}\\big)^\\gamma, \\quad\\text{for all } u>0.\n\t\\]\n\tThen\n\t\\[\n\t\t\\lim_{t\\to+\\infty}\n\t\t\\sup_{g \\in \\mathcal{G}}\n\t\t\\bigg| \\int_0^{\\infty}g(u) \\: \\mu_t({\\rm d} u) - \\int_0^{\\infty}g(u)\\: \\mu({\\rm d} u)\\bigg|=0.\n\\]\n\\end{lemma}\n\\begin{proof}\n\tLet $\\epsilon \\in(0,1)$. By \\eqref{ineq:sup-finite}, for sufficiently large $M > 1$, we have\n\t\\[\n\t\t\\int_{(0,1/M] \\cup [M,\\infty)}\n\t\tg(u) \\: \\mu_t({\\rm d} u)\n\t\t\\leq\n\t\t\\frac{C}{ M+ M^{-1}}\n\t\t\\sup_{t \\geq 1} \\int_0^{\\infty} \\big(u+u^{-1}\\big)^{\\gamma+1} \\mu_t({\\rm d} u) \\leq \\epsilon/3.\n\t\\]\n\tA similar inequality holds true for $\\mu$ in place of $\\mu_t$. Notice that $M$ may be chosen so that the inequality holds\n\ttrue uniformly for all $g \\in \\mathcal{G}$ and $t \\geq 1$. Next we write\n\t\\begin{align*}\n\t\t&\\bigg| \\int_{(1/M, M)} g(u) \\: \\mu_t({\\rm d} u) - \\int_{(1/M,M)} g(u) \\: \\mu({\\rm d} u) \\bigg| \\\\\n\t\t&\\qquad\n\t\t\\leq \\int_{(1/M,M)} g(u) |\\mu_t-\\mu| ({\\rm d} u) \\\\\n\t\t&\\qquad= C \\big(M+ M^{-1}\\big)^{\\gamma} 2 \\|\\mu_t-\\mu\\|_{TV}\n\t\\end{align*}\n\twhere $|\\mu_t-\\mu|({\\rm d}u)$ denotes the total variation measure. Finally, by Lemma~\\ref{lem:mu_t_erg}, there is $T \\geq 1$\n\tsuch that for all $t\\geq T$ we have\n\t\\[\n\t\tC \\big(M+ M^{-1}\\big)^{\\gamma} 2 \\|\\mu_t-\\mu\\|_{TV} \\leq \\epsilon/3\n\t\\]\n\twhich completes the proof.\n\\end{proof}\n\nGiven $c \\in (0, 1)$, for $\\delta > 0$ and $t > 0$, we set\n\\begin{equation}\n\t\\label{eq:52}\n\t\\mathscr{X}(t; \\delta) = \\bigg\\{x \\in \\RR^d : -\\log_c \\norm{x} \\leq \\frac{t}{1+\\delta} \\bigg\\}.\n\\end{equation}\n\n\\begin{lemma}\n\t\\label{lem:unif_conv-2}\n\tLet $c \\in (0, 1)$, $\\alpha \\in (0, 2]$ and $m = c^\\alpha$. Fix $n \\in \\NN$ and $C, \\gamma > 0$.\n\tLet $\\mathcal{F}$ denote the family of functions $f : (0,\\infty)\\times \\RR^n \\to \\RR$, such that for all $x \\in \\RR^n$\n\tand $u>0$,\n\t\\begin{align}\n\t\t\\label{eq:78}\n\t\t0 \\leq f(u,x) \\leq C \\big(u+u^{-1} \\big)^\\gamma,\\qquad\n\t\t| f(u, x)-f(u,0)|\\leq C |x| \\big(u+u^{-1} \\big)^\\gamma.\n\t\\end{align}\n\tThen for each $\\delta > 0$, we have\n\t\\begin{align*}\n\t\t\\lim_{t \\to +\\infty}\n\t\t\\sup_{\\stackrel{x \\in \\mathscr{X}(t; \\delta)}{f \\in \\mathcal{F}}}\n\t\t\\bigg| e^{-t}f(t,x)+e^{-t}\\sum_{j=1}^\\infty t^j \\int_0^1 f(tu,c^jx) P_j(u) \\: {\\rm d}u\n\t\t-\\int_0^{\\infty} f(u,0) \\: \\mu({\\rm d} u) \\bigg|=0.\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}\n\tIn view of Lemma \\ref{lem:unif_conv-1}, it is enough to show that\n\t\\begin{equation}\n\t\t\\lim_{t \\to +\\infty} \\sup_{\\stackrel{x \\in \\mathscr{X}(t; \\delta)}{f \\in \\mathcal{F}}}\n\t\t\\bigg|\n\t\te^{-t} f(t, x) + e^{-t} \\sum_{j = 1}^\\infty t^j \\int_0^1 f(tu, c^j x) P_j(u) {\\: \\rm d} u\n\t\t-\n\t\t\\int_0^\\infty f(u, 0) \\mu_t({\\rm d} u)\n\t\t\\bigg|=0.\n\t\\end{equation}\n\tGiven $\\epsilon > 0$ we set\n\t\\[\n\t\tN = \\max\\big\\{\\gamma + 1, -\\log_c (\\norm{x}/\\epsilon ) \\big\\}.\n\t\\]\n\tBy \\eqref{def:mu_t}, we can write\n\t\\begin{align*}\n\t\t&\n\t\t\\bigg|\n\t\te^{-t} f(t, x) + e^{-t} \\sum_{j = 1}^\\infty t^j \\int_0^1 f(tu, c^j x) P_j(u) {\\: \\rm d} u\n\t\t- \\int_0^\\infty f(u, 0) {\\: \\rm d} \\mu_t({\\rm d} u) \\bigg|\\\\\n\t\t&\\qquad\\qquad\n\t\t\\leq\n\t\te^{-t} \\big( \\abs{f(t, x)} + \\abs{f(t, 0)}\\big)\n\t\t+ e^{-t} \\sum_{1 \\leq j \\leq N} t^j \\int_0^1 \\big(\\abs{f(tu, c^j x)} + \\abs{f(tu, 0)} \\big) P_j(u) {\\: \\rm d} u \\\\\n\t\t&\\qquad\\qquad\\phantom{\\leq}\n\t\t+ e^{-t} \\sum_{j > N} t^j \\int_0^1 \\big| f(tu, c^j x) - f(tu, 0) \\big| P_j(u) {\\: \\rm d} u.\n\t\\end{align*}\n\tFirst, let us observe that if $x \\in \\mathscr{X}(t; \\delta)$, then for $j > N$,\n\t$c^j \\norm{x} \\leq c^N \\norm{x} \\leq \\epsilon$, and thus\n\t\\begin{align*}\n\t\te^{-t} \\sum_{j > N} t^j \\int_0^1 \\big| f(tu, c^j x) - f(tu, 0) \\big| P_j(u) {\\: \\rm d} u\n\t\t&\\leq\n\t\tC e^{-t}\n\t\t\\sum_{j > N} t^j \\int_0^1 c^j\\norm{x} \\Big( tu+\\frac1{tu}\\Big)^\\gamma P_j(u) \\: {\\rm d}u \\\\\n\t\t&\\leq\n\t\t\\epsilon\n\t\tC \\int_0^\\infty \\big(u + u^{-1}\\big)^\\gamma \\mu_t({\\rm d} u)\n\t\\end{align*}\n\twhich by Proposition \\ref{prop:m-1b} is uniformly bounded by a constant multiply of $\\epsilon$. Next, by\n\tTheorem \\ref{thm:all-moments}, for each $j \\in \\NN$,\n\t\\begin{align}\n\t\t\\label{eq:56}\n\t\t\\int_0^1 \\abs{f(tu,c^jx)} P_j(u) \\: {\\rm d}u\n\t\t\\leq\n\t\tC\n\t\t\\frac{2^{\\lceil \\gamma \\rceil - 1}}{(m;m)_\\infty}\n\t\t&\\bigg(\n\t\tt^{\\lceil \\gamma \\rceil}\n\t\t\\bigg\\{\\prod_{k=1}^j \\frac{1-m^{k+\\lceil \\gamma \\rceil}}{k+\\lceil \\gamma \\rceil}\\bigg\\}\n\t\t+ t^{-\\lceil \\gamma \\rceil}\n\t\t\\bigg\\{\\prod_{k=1}^j \\frac{1-m^{k-\\lceil \\gamma \\rceil}}{k-\\lceil \\gamma \\rceil}\\bigg\\}\n\t\t\\bigg).\n\t\\end{align}\n\tIf $j \\geq \\lceil \\gamma \\rceil$, the second product is understood in the limiting sense, that is\n\t\\[\n\t\t\\prod_{k=1}^j \\frac{1-m^{k-\\lceil \\gamma \\rceil}}{k-\\lceil \\gamma \\rceil}\n\t\t=\n\t\t(-\\log m)\n\t\t\\prod_{\\stackrel{k=1}{k \\neq \\lceil \\gamma \\rceil}}^j \\frac{1-m^{k-\\lceil \\gamma \\rceil}}{k-\\lceil \\gamma \\rceil}.\n\t\\]\n\tNow, using \\eqref{eq:56}, we get\n\t\\[\n\t\t\\sum_{1 \\leq j \\leq N} t^j \\int_0^1 \\abs{f(tu, c^j x)} P_j(u) {\\: \\rm d} u\n\t\t\\leq\n\t\tC \\frac{2^{\\lceil \\gamma \\rceil-1}}{(m; m)_\\infty} \\big(I_1 + I_2\\big)\n\t\\]\n\twhere\n\t\\[\n\t\tI_1=\\sum_{1 \\leq j \\leq N} t^{j+\\lceil \\gamma \\rceil}\n\t\t\\bigg\\{\\prod_{k=1}^j \\frac{1-m^{k+\\lceil \\gamma \\rceil}}{k+\\lceil \\gamma \\rceil}\\bigg\\},\n\t\t\\quad\\text{and}\\quad\n\t\tI_2 = \\sum_{1 \\leq j \\leq N} t^{j-\\lceil \\gamma \\rceil}\n\t\t\\bigg\\{\\prod_{k=1}^j \\frac{1-m^{k-\\lceil \\gamma \\rceil}}{k-\\lceil \\gamma \\rceil}\\bigg\\}.\n\t\\]\n\tThere is $c_1 > 0$ such that\n\t\\[\n\t\tI_1 \\leq\n\t\tc_1 \\sum_{1 \\leq j \\leq N}\n\t\t\\frac{t^{j+\\lceil \\gamma \\rceil}}{(j+\\lceil \\gamma \\rceil)!}\n\t\t\\leq\n\t\tc_1 S(t, x)\n\t\\]\n\twhere\n\t\\[\n\t\tS(t, x)= \\sum_{0 \\leq j \\leq N+\\lceil \\gamma \\rceil}\\frac{t^j}{j!}.\n\t\\]\n\tTo bound $I_2$ for $t \\geq 1$, we write\n\t\\begin{align*}\n\t\tI_2\n\t\t&\\leq \\sum_{j = 1}^{\\lceil \\gamma \\rceil-1} t^{j-\\lceil \\gamma \\rceil}\n\t\t\\bigg\\{\n\t\t\\prod_{k=1}^j \\frac{1-m^{k-\\lceil \\gamma \\rceil}}{k-\\lceil \\gamma \\rceil}\\bigg\\}\n\t\t+\n\t\t(-\\log m)\n\t\t\\sum_{\\lceil \\gamma \\rceil \\leq j \\leq N}\n\t\tt^{j-\\lceil \\gamma \\rceil}\n\t\t\\bigg\\{\n\t\t\\prod_{\\stackrel{k=1}{k \\neq \\lceil \\gamma \\rceil}}^j\n\t\t\\frac{1-m^{k-\\lceil \\gamma \\rceil}}{k-\\lceil \\gamma \\rceil}\n\t\t\\bigg\\} \\\\\n\t\t&\\leq\n\t\tc_2 + c_3 \\sum_{0 \\leq j \\leq N-\\lceil \\gamma \\rceil}\n\t\t\\frac{t^j}{j!}\n\t\\end{align*}\n\tfor certain $c_2, c_3 > 0$. Hence, there is $c_4 > 0$ such that\n\t\\[\n\t\t\\abs{f(t, x)} + \\sum_{1 \\leq j \\leq N} t^j \\int_0^1 \\abs{f(tu, c^j x)} P_j(u) {\\: \\rm d} u\n\t\t\\leq\n\t\tc_4 S(t, x).\n\t\\]\n\tIt remains to show that\n\t\\begin{equation}\n\t\t\\label{eq:50}\n\t\t\\lim_{t \\to +\\infty} e^{-t} \\sup_{x \\in \\mathscr{X}(t; \\delta)} S(t, x) = 0.\n\t\\end{equation}\n\tLet $T = \\lceil N + \\gamma \\rceil$. If $-\\log_c (\\norm{x}/\\epsilon) \\leq \\gamma+1$, then $N = \\gamma+1$, and so for\n\t$t > T$,\n\t\\[\n\t\tS(t, x) \\leq e t^{2\\gamma+1}.\n\t\\]\n\tOtherwise, $N = - \\log_c (\\norm{x}/\\epsilon)$. Since $x \\in \\mathscr{X}(t; \\delta)$, if\n\t\\[\n\t\tt \\geq \\frac{(1+\\delta)^2}{\\delta} (\\log_c \\epsilon + \\gamma + 1),\n\t\\]\n\twe have $t > (1+\\delta)T$, and so\n\t\\[\n\t\tS(t; x) = \\sum_{j = 0}^T \\frac{t^j}{j!} \\leq (T+1) \\frac{t^T}{T!}.\n\t\\]\n\tThe function $[1, \\infty) \\ni s \\mapsto s^{-1} \\log s$ is decreasing, hence for $t \\geq (1+\\delta) T$, we get\n\twe have\n\t\\[\n\t\tT \\log \\frac{t}{T} \\leq \\frac{\\log (1+\\delta)}{1+\\delta} t.\n\t\\]\n\tNow, by the Stirling's formula\n\t\\begin{align*}\n\t\tT \\frac{t^T}{T!} e^{-t}\n\t\t&\\leq\n\t\tc_5\n\t\t\\sqrt{T}\n\t\t\\exp\\big\\{ T \\log t - T \\log T + T - t\\big\\} \\\\\n\t\t&\\leq\n\t\tc_5 \\sqrt{\\frac{t}{1+\\delta}} \\exp\\Big\\{t \\frac{\\log(1+\\delta)-\\delta}{1+\\delta}\\Big\\},\n\t\\end{align*}\n\tand because\n\t\\[\n\t\t\\frac{\\log(1+\\delta)-\\delta}{1+\\delta} < 0,\n\t\\]\n\twe obtain \\eqref{eq:50} and the theorem follows.\n\\end{proof}\n\n\\subsection{The function $\\rho_{\\mathbf{Y}}$}\n\\label{sec:2.4}\nLet $(\\mu_t : t > 0)$ be the family of probability measures defined by \\eqref{def:mu_t} for certain $c \\in (0, 1)$. In view\nof Theorem \\ref{thm:weak_conv}, $\\mu_t$ weakly converge to the probability measure $\\mu$ as $t$ tends to infinity. For a\ngiven strictly $\\alpha$-stable process $\\mathbf{Y}$ in $\\RR^d$, $\\alpha \\in (0, 2]$, with a transition density $p_0$, we\ndefine a function\n\\begin{equation}\n\t\\label{representationrho}\n\t\\rho_{\\mathbf{Y}} (y) = \\int_0^{\\infty} p_0(u;0,y) \\: \\mu({\\rm d} u), \\qquad y \\in \\RR^d.\n\\end{equation}\nIn this section we investigate the properties of $\\rho_{\\mathbf{Y}}$.\n\\begin{proposition}\n\t\\label{prop:6}\n\tSuppose that $\\mathbf{Y}$ is a strictly $\\alpha$-stable process in $\\RR^d$, $\\alpha \\in (0, 2]$, with a transition density.\n\tThen $\\rho_{\\mathbf{Y}} \\in \\calC_0^\\infty(\\RR^d)$. Moreover, for all $\\gamma \\in \\RR$,\n\t\\begin{equation}\n\t\t\\label{eq:57}\n\t\t\\int_{\\RR^d} |y|^{\\gamma} \\rho_{\\mathbf{Y}}(y) {\\: \\rm d}y\n\t\t= \\frac{\\Gamma(\\gamma/\\alpha+1)}{\\Gamma_m(\\gamma/\\alpha+1)} (1-m)^{-\\gamma/\\alpha}\\, \\mathbb{E}|Y_1|^\\gamma.\n\t\\end{equation}\n\\end{proposition}\n\\begin{proof}\n\tThe regularity of $\\rho_{\\mathbf{Y}}$ follows by Lemma~\\ref{lem:A3}\\eqref{en:3:2}, and the finiteness of all moments of\n\t$\\mu$, see Theorem~\\ref{thm:weak_conv}. Next, by the scaling Lemma \\ref{lem:A3}\\eqref{en:3:1} and Tonelli's theorem, we get\n\t\\begin{align*}\n\t\t\\int_{\\RR^d} |y|^{\\gamma} \\rho_{\\mathbf{Y}}(y) {\\: \\rm d}y\n\t\t&= \\int_0^\\infty u^{-d/\\alpha} \\int_{\\RR^d}|y|^\\gamma p_0(1;0, u^{-1/\\alpha} y){\\: \\rm d} y \\: \\mu({\\rm d}u)\\\\\n\t\t&=\\int_0^\\infty u^{\\gamma/\\alpha}\\mu({\\rm d}u) \\int_{\\RR^d}|z|^\\gamma p_0(1;0, z){\\: \\rm d} z,\n\t\\end{align*}\n\twhich in view of \\eqref{eq:55} completes the proof.\n\\end{proof}\n\n\\begin{remark}\n\tLet us recall that a function $f: \\RR^d \\to \\RR$ is a homogeneous function of degree $\\gamma \\in \\RR$, if\n\t$f(sx)=s^\\gamma f(x)$ for all $x\\in\\RR^d$ and $s > 0$. Reasoning similar to the proof of \\eqref{eq:57} leads to\n\tthe following statement: Under the assumptions of Proposition \\ref{prop:6}, let $f$ be a homogeneous function of degree $\\gamma \\in \\RR$, such that\n\t\\[\n\t\t\\sup_{|\\theta|=1} |f(\\theta)|<\\infty.\n\t\\]\n\tIf\n\t\\[\n\t\t\\gamma >\n\t\t\\begin{cases}\n\t\t\t-\\frac{\\alpha}{d} & \\text{if } \\alpha \\in (0, 2), \\\\\n\t\t\t-d & \\text{if } \\alpha = 2,\n\t\t\\end{cases}\n\t\\]\n\tthen\n\t\\[\n\t\t\\int_{\\RR^d} f(y) \\rho_{\\mathbf{Y}}(y){\\: \\rm d}y =\n\t\t\\frac{\\Gamma(\\gamma/\\alpha+1)}{\\Gamma_m(\\gamma/\\alpha+1)}\n\t\t(1-m)^{-\\gamma/\\alpha} \\: \\mathbb{E} f(Y_1)\n\t\\]\n\tand the integral converges absolutely.\n\\end{remark}\n\nThere are examples of strictly $\\alpha$-stable processes in $\\RR$ for which the explicit expression for all moments is\nknown, see e.g. \\cite[Proposition 1.4]{Hardin}.\n\\begin{example}[{see \\cite{Shanbhag} and \\cite[Example 25.10]{MR1739520}}]\n\tIf $d=1$ and $\\mathbf{Y}$ is an $\\alpha$-stable subordinator with $\\alpha \\in (0,1)$, then for $\\gamma<\\alpha$,\n\t\\[\n\t\t\\mathbb{E}|Y_1|^\\gamma= \\frac{\\Gamma\\left(1-\\frac{\\gamma}{\\alpha}\\right)}{\\Gamma\\left(1-\\gamma\\right)}.\n\t\\]\n\\end{example}\n\n\\begin{example}[see \\cite{Shanbhag}]\n\tIf $d = 1$ and $\\mathbf{Y}$ is a symmetric $\\alpha$-stable process with $\\alpha \\in (0,2)$,\n\tthen for $-1<\\gamma<\\alpha$,\n\t\\[\n\t\t\\mathbb{E}|Y_1|^\\gamma\n\t\t=\\frac{2^\\gamma \\Gamma (\\tfrac{1+\\gamma}{2})\\Gamma (1-\\tfrac{\\gamma}{\\alpha})}{\\sqrt{\\pi}\\,\\Gamma(1-\\tfrac{\\gamma}{2})}.\n\t\\]\n\\end{example}\n\n\\begin{example}\n\tIf $d = 1$ and $\\mathbf{Y}$ is a Brownian motion, then for $\\gamma>-1$ one can see that\n\t\\[\n\t\t\\mathbb{E}|Y_1|^\\gamma=\\frac{2^{\\gamma}\\Gamma \\big(\\tfrac{1+\\gamma}{2}\\big)}{\\sqrt{\\pi}}.\n\t\\]\n\tIn particular, by \\eqref{eq:57} for $k \\in \\NN$,\n\t\\[\n\t\t\\int_{\\RR} y^{2k-1} \\rho_{\\mathbf{Y}}(y){\\: \\rm d}y=0, \\quad\\text{ and }\\quad\n\t\t\\int_{\\RR} y^{2k} \\rho_{\\mathbf{Y}}(y){\\: \\rm d}y= \\frac{(2k)!}{(m;m)_k}.\n\t\\]\n\\end{example}\n\n\\begin{lemma}\n\t\\label{lem:densities}\n\tSuppose that $\\mathbf{Y}$ is a strictly $\\alpha$-stable process in $\\RR^d$, $\\alpha \\in (0, 2]$, with a transition density.\n\tThen, for all $y \\in \\RR^d$,\n\t\\begin{equation}\n\t\t\\label{eq:66}\n\t\t\\rho_{\\mathbf{Y}}(y)\n\t\t=\\frac{1}{(m; m)_\\infty}\n\t\t\\sum_{k=0}^\\infty (-1)^k \\frac{m^{\\frac{1}{2}k(k-1)}}{(m; m)_k} \\, U^{(m^{-k})}(y)\n\t\\end{equation}\n\twhere $U^{(\\beta)}(y)=\\int_0^\\infty e^{-\\beta u} \\,p_0(u;0,y) {\\: \\rm d}u$, $\\beta > 0$. Moreover,\nif $\\mathbf{Y}$ is a Brownian motion in $\\RR^d$, then\n\t\\begin{equation}\n\t\t\\label{eq:67}\n\t\t\\lim_{\\norm{y} \\to +\\infty}\n\t\t\\frac{\\rho_{\\mathbf{Y}}(y)}{|y|^{-\\frac{d-1}{2}}e^{-|y|}}\n\t\t= \\frac12 \\frac{1}{(m; m)_\\infty} (2\\pi)^{-\\frac{d-1}{2}}.\n\t\\end{equation}\n\\end{lemma}\n\\begin{proof}\n\tThe formula follows immediately from \\eqref{representationrho} and \\eqref{eq:81}. Indeed, we get\n\t\\[\n\t\t\\rho_{\\mathbf{Y}}(y) = \\int_0^\\infty p_0(u; 0, y) \\mu({\\rm d} u)\n\t\t=\n\t\t\\frac{1}{(m; m)_\\infty} \\sum_{k = 0}^\\infty \\frac{m^{\\frac{1}{2}k(k-1)}}{(m; m)_k} \\int_0^\\infty\n\t\te^{-m^k u} p_0(u; 0, y) \\mu({\\rm d} u).\n\t\\]\n\tTo show \\eqref{eq:67}, let us recall that\n\t\\begin{equation}\n\t\t\\label{eq:U_Bessel}\n\t\tU^{(\\beta)}(y)=(2\\pi)^{-d/2}\\left(\\frac{|y|}{\\sqrt{\\beta}}\\right)^{1-d/2} K_{d/2-1}(\\sqrt{\\beta}|y|)\n\t\\end{equation}\n\twhere $K_{d/2-1}$ is the modified Bessel function of the second type. Hence, by \\eqref{eq:66} we get\n\t\\[\n\t\t\\frac{\\rho_{\\mathbf{Y}}(y)}{|y|^{-\\frac{d-1}{2}}e^{-|y|}}\n\t\t=\n\t\t\\frac{1}{(m; m)_\\infty}\n\t\t\\left( \\frac{U^{(1)}(y)}{{|y|^{-\\frac{d-1}{2}}e^{-|y|}}}\\right)\n\t\t\\sum_{k=0}^\\infty (-1)^k \\frac{m^{\\frac{1}{2}k(k-1)}}{(m; m)_k} \\frac{U^{(m^{-k})}(y)}{U^{(1)}(y)}.\n\t\\]\n\tBecause\n\t\\[\n\t\t\\lim_{|y|\\to +\\infty}\n\t\t\\left(\\frac{U^{(1)}(y)}{{|y|^{-\\frac{d-1}{2}}e^{-|y|}}}\\right)\n\t\t= \\frac12 (2\\pi)^{-\\frac{d-1}{2}},\n\t\\]\n\tand\n\t\\[\n\t\t0\\leq \\frac{U^{(m^{-k})}(y)}{U^{(1)}(y)}\\leq 1\n\t\\]\n\tit is enough to treat sums of the form\n\t\\[\n\t\t1+ \\sum_{k=1}^N (-1)^k \\frac{m^{\\frac{1}{2}k(k-1)}}{(m; m)_k} \\frac{U^{(m^{-k})}(y)}{U^{(1)}(y)}\n\t\\]\n\tfor a fixed $N$. For each $k \\in \\{1, \\ldots, N\\}$, by \\eqref{eq:U_Bessel} and the asymptotic behavior of $K_{d/2-1}$,\n\twe obtain\n\t\\[\n\t\t\\lim_{|y|\\to +\\infty} \\frac{U^{(m^{-k})}(y)}{U^{(1)}(y)}=0,\n\t\\]\n\twhich completes the proof.\n\\end{proof}\n\n\\begin{remark}\n\tUnder the assumptions of Lemma \\ref{lem:densities} for $d = 1$, we get the following formula\n\t\\[\n\t\t\\rho_{\\mathbf{Y}} (y)= \\frac12 \\frac{1}{(m; m)_\\infty}\\sum_{k=0}^\\infty (-1)^k\n\t\t\\frac{m^{\\frac{k^2}{2}}}{(m; m)_k}\n\t\te^{-m^{-\\frac{k}2 }|y|},\n\t\\]\n\tand\n\t\\[\n\t\t\\lim_{|y|\\to +\\infty} \\rho_{\\mathbf{Y}}(y) \\,e^{|y|} = \\frac12 \\frac{1}{(m; m)_\\infty}.\n\t\\]\n\\end{remark}\n\n\\section{Stable processes with resetting}\n\\label{sec:stationary}\nIn this section, we describe the resetting procedure for a large class of L{\\'e}vy processes.\nLet $\\mathbf{Y}$ be a L{\\'e}vy process in $\\RR^d$ and let $c \\in (0, 1)$.\nWe consider another L\\'{e}vy process $\\big((Y_t, N_t) : t \\geq 0\\big)$ with a natural filtration\n$\\big(\\tilde{\\calF}_t : t \\geq 0\\big)$ where $\\mathbf{N}$ is a Poisson process independent of $\\mathbf{Y}$ with intensity $1$.\nThen the Poisson arrival moments $(T_k : k \\geq 0)$ are Markov times relative to $(\\tilde{\\calF}_t : t\\geq 0)$ where $T_0=0$.\nLet $\\mathbf{X}$ be a process with resetting defined in \\eqref{eq:18}. Then for all $t > 0$ and every Borel set\n$A \\subset \\RR^d$, we have\n\\begin{align*}\n\t&\n\t\\PP^{(x,0)} (X_t \\in A) \\\\\n\t&\\qquad= \\sum^\\infty_{n=0} \\PP^{(x, 0)}( X_t\\in A, N_t=n) \\\\\n\t&\\qquad= \\sum^\\infty_{n=0}\n\t\\PP^{(x,0)} \\bigg(\\sum^n_{k=0} c^{n-k} \\Big(Y_{t \\wedge T_{k+1}}-Y_{T_k}\\Big)+c^n x \\in A, T_n \\leq t <T_{n+1}\\bigg)\\\\\n\t&\\qquad= \\PP^{(x,0)}(Y_t \\in A) \\PP^{(x, 0)}(N_t=0) +\n\t\\sum^\\infty_{n=1} \\PP^{(x,0)}\\bigg(\\sum^n_{k=1}c^{n-k}\n\t\\Big(Y_{t\\wedge T_{k+1}}-Y_{T_k}\\Big)+c^nY_{T_1}\\in A,T_n\\leq t <T_{n+1}\\bigg).\n\\end{align*}\nHence, if for each $t > 0$, $Y_t$ has an absolutely continuous distribution, then for each $t > 0$ also $X_t$ has an absolutely\ncontinuous distribution. Let us denote by $p_0$ and $p$ the transition densities of $\\mathbf{Y}$ and $\\mathbf{X}$, respectively.\nUsing the strong Markov property we get $\\big( (\\tilde{Y}_t, \\tilde{N}_t) : t \\geq 0\\big)\n= \\big((Y_{t+T_1}-Y_{T_1}, N_{t+T_1}-1) : t \\geq 0\\big)$ is a L\\'evy process with the same distribution as\n$\\big((Y_t,N_t) : t \\geq 0\\big)$ and independent of $Y_{T_1}$. Hence,\n\\begin{equation}\n\t\\label{eq:pomoc}\n\t\\begin{aligned}\n\t\\PP^{(x,0)}(X_t\\in A)\n\t&= \\PP^{(x,0)}(Y_t\\in A)e^{-t}\\\\\n\t&\\phantom{=}\n\t+\n\t\\sum^\\infty_{n=1} \\PP^{(x,0)}\\bigg(\\sum^{n-1}_{k=0}c^{n-(k+1)}\n\t\\Big(\\tilde{Y}_{(t-T_1) \\wedge \\tilde{T}_{k+1}}-\\tilde{Y}_{\\tilde{T}_k}\\Big)\n\t+c^nY_{T_1}\\in A,\\tilde{T}_{n-1}\\leq t-T_1 <\\tilde{T}_{n}\\bigg)\\\\\n\t&=\\PP^{(x,0)}(Y_t\\in A)e^{-t}+\\PP^{(x,0)}\\Big(T_1\\leq t,\\PP^{(cY_{T_1},0)}(\\tilde{X}_{t-T_1}\\in A)\\Big),\n\\end{aligned}\n\\end{equation}\nthat is the transition densities satisfy\n\\begin{equation}\n\t\\label{eq:6}\n\tp(t; x, y)\n\t=\n\te^{-t} p_0(t; x, y)\n\t+\n\t\\int_0^t \\int_{\\RR^d}\n\te^{-s} p_0(s; x, z) p(t-s; cz, y) {\\: \\rm d} z {\\: \\rm d} s,\n\t\\qquad x, y \\in \\RR^d, t > 0.\n\\end{equation}\nSimilarly, the natural filtration of $\\textbf{X}$ is a subfiltration of $(\\tilde{\\calF}_t : t \\geq 0)$, thus\n$\\PP^{(x,0)}(X_{t+u}\\in A | \\tilde{\\calF_t})$ is a function of $X_t$, that is $\\mathbf{X}$ is a Markov process.\n\\begin{proposition}\n\t\\label{prop:5}\n\tSuppose that $\\mathbf{Y}$ is a L{\\'e}vy process in $\\RR^d$ with a transition density $p_0$.\n\tAssume that $\\mathbf{X}$ is obtained from $\\mathbf{Y}$ by partial resetting with factor $c\\in(0,1)$. Then the process\n\t$\\mathbf{X}$ has the transition density $p$ satisfying\n\t\\begin{equation}\n\t\t\\label{eq:2}\n\t\tp(t; x, y) = e^{-t} \\sum_{j = 0}^\\infty p_j(t; x, y), \\quad \\text{for all } x,y \\in \\RR^d, t > 0\n\t\\end{equation}\n\twhere $(p_n : n \\in \\NN_0)$ is a sequence of functions that satisfy the recursion\n\t\\begin{equation}\n\t\t\\label{eq:15}\n\t\tp_{n+1}(t; x, y) = \\int_0^t \\int_{\\RR^d} p_0(s; x, z) p_n(t-s; cz, y) {\\: \\rm d}z {\\: \\rm d} s,\n\t\t\\quad\\text{for all }x, y \\in \\RR^d, t >0.\n\t\\end{equation}\n\\end{proposition}\n\\begin{proof}\n\tFirst, let us observe that for all $j \\in \\NN_0$, $x \\in \\RR^d$ and $t > 0$,\n\t\\begin{equation}\n\t\t\\label{eq:20}\n\t\t\\int_{\\RR^d} p_j(t; x, y) {\\: \\rm d} y = \\frac{t^j}{j!}.\n\t\\end{equation}\n\tTo see this, we reason by induction over $j \\in \\NN_0$. For $j = 0$ the formula trivially holds true. Next,\n\t\\begin{align*}\n\t\t\\int_{\\RR^d} p_{j+1}(t; x, y) {\\: \\rm d} y\n\t\t&=\n\t\t\\int_{\\RR^d} \\int_0^t \\int_{\\RR^d} p_0(s; x, z) p_j(t-s; cz, y) {\\: \\rm d} z {\\: \\rm d} s {\\: \\rm d} y \\\\\n\t\t&=\n\t\t\\frac{1}{j!}\n\t\t\\int_0^t \\int_{\\RR^d} p_0(s; x, z) (t-s)^j {\\: \\rm d}z {\\: \\rm d} s \\\\\n\t\t&=\n\t\t\\frac{1}{j!}\n\t\t\\int_0^t (t-s)^j {\\: \\rm d} s = \\frac{t^{j+1}}{(j+1)!},\n\t\\end{align*}\n\tas claimed.\n\n\tSince all the terms of the series \\eqref{eq:2} are positive, by Tonelli's theorem \\eqref{eq:20} we get\n\t\\begin{align*}\n\t\t\\int_{\\RR^d} e^{-t} \\sum_{j = 0}^\\infty p_j(t; x, y) {\\: \\rm d} y\n\t\t&=\n\t\te^{-t} \\sum_{j = 0}^\\infty \\int_{\\RR^d} p_j(t; x, y) {\\: \\rm d} y \\\\\n\t\t&=\n\t\te^{-t} \\sum_{j = 0}^\\infty \\frac{t^j}{j!} = 1.\n\t\\end{align*}\n\tHence, \\eqref{eq:2} defines actual solution of \\eqref{eq:6}. Next, we observe that it is also the unique probabilistic\n\tsolution of \\eqref{eq:6}. Indeed, if $\\tilde{p}(t; x , y)$ would be another solution, then\n\t\\[\n\t\te^{-t} p_0(t; x, y) \\leq \\tilde{p}(t; x, y), \\quad \\text{for all } x, y \\in \\RR^d, t > 0.\n\t\\]\n\tThus reasoning by induction one can prove that for all $N \\in \\NN$,\n\t\\[\n\t\te^{-t} \\sum_{j = 0}^N p_j(t; x, y) \\leq \\tilde{p}(t; x, y), \\quad \\text{for all } x, y \\in \\RR^d, t > 0.\n\t\\]\n\tConsequently,\n\t\\[\n\t\tp(t; x, y) \\leq \\tilde{p}(t; x, y), \\quad \\text{for all } x, y \\in \\RR^d, t > 0.\n\t\\]\n\tSince both $p(t; x, \\cdot)$ and $\\tilde{p}(t; x, \\cdot)$ integrate to $1$, for each $\\epsilon > 0$\n\tby Chebyshev's inequality we obtain\n\t\\[\n\t\t\\big|\\big\\{y \\in \\RR^d : p(t; x, y) \\geq \\tilde{p}(t; x, y) + \\epsilon \\big\\}\\big|\n\t\t\\leq\n\t\t\\epsilon^{-1} \\int_{\\RR^d} (p(t; x, y) - \\tilde{p}(t; x, y)) {\\: \\rm d} y = 0,\n\t\\]\n\tthat is $p(t; x, y) = \\tilde{p}(t; x, y)$. This completes the proof.\n\\end{proof}\n\nLet us observe that certain properties of $p_0$ translate to $p_n$.\n\\begin{proposition}\n\t\\label{prop:4}\n\tSuppose that $\\mathbf{Y}$ is a L\\'evy process on $\\RR^d$ with a transition density $p_0$.\n\t\\begin{enumerate}[label=\\rm (\\roman*), start=1, ref=\\roman*]\n\t\t\\item\n\t\t\\label{en:2:1}\n\t\tThen for all $n \\in \\NN$,\n\t\t\\[\n\t\t\tp_n(t; x, y) = p_n(t; 0, y - c^n x).\n\t\t\\]\n\t\t\\item\n\t\t\\label{en:2:2}\n\t\tIf there is $\\Upsilon \\in \\GL(\\RR, d)$ such that\n\t\t\\[\n\t\t\tp_0(t; x, y) = p_0(t; \\Upsilon x, \\Upsilon y), \\quad \\text{for all } x, y \\in \\RR^d, t > 0,\n\t\t\\]\n\t\tthen for each $j \\in \\NN$,\n\t\t\\[\n\t\t\tp_j(t; x, y) = p_j(t; \\Upsilon x, \\Upsilon y), \\quad \\text{for all } x, y \\in \\RR^d, t > 0.\n\t\t\\]\n\t\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\n\tThe proof of \\eqref{en:2:1} is by induction with respect to $n \\in \\NN$. For $n = 0$ the formula trivially holds true.\n\tUsing the inductive hypothesis and the change of variables we can write\n\t\\begin{align*}\n\t\tp_{n+1}(t; x, y)\n\t\t&= \\int_0^t \\int_{\\RR^d} p_0(s; x, z) p_n(t-s; cz, y) {\\: \\rm d}z {\\: \\rm d} s\\\\\n\t\t&= \\int_0^t \\int_{\\RR^d} p_0(s; 0, z-x) p_n(t-s; cz, y) {\\: \\rm d} z {\\: \\rm d} s \\\\\n\t\t&= \\int_0^t \\int_{\\RR^d} p_0(s; 0, z) p_n(t-s; c(z +x), y) {\\: \\rm d} z {\\: \\rm d} s \\\\\n\t\t&= \\int_0^t \\int_{\\RR^d} p_0(s; 0, z) p_n(t-s; cz , y - c^{n+1} x) {\\: \\rm d} z {\\: \\rm d} s \\\\\n\t\t&= p_{n+1}(t; 0, y - c^{n+1} x),\n\t\\end{align*}\n\tcompleting the proof of the first part of the proposition. To show \\eqref{en:2:2} we again use induction to get\n\t\\begin{align*}\n\t\tp_{j+1}(t; \\Upsilon x, \\Upsilon y)\n\t\t&= \\int_0^t \\int_{\\RR^d} p_0(s; \\Upsilon x, z) p_j(t-s; cz, \\Upsilon y) {\\: \\rm d}z {\\: \\rm d} s \\\\\n\t\t&= \\int_0^t \\int_{\\RR^d} p_0(s; x, \\Upsilon^{-1} z) p_j(t-s; c\\Upsilon^{-1} z, y) (\\det \\Upsilon)^{j}\n\t\t{\\: \\rm d}z {\\: \\rm d} s \\\\\n\t\t&= (\\det \\Upsilon)^{j+1} p_{j+1}(t; x, y).\n\t\\end{align*}\n\tNote that $p(t; x, y)$ integrates to $1$, hence $\\det \\Upsilon =1$ and the proposition follows.\n\\end{proof}\n\n\\begin{remark}\n\tDue to Proposition \\ref{prop:4}\\eqref{en:2:2} we can easily see that $p_n$ is isotropic if $p_0$ is isotropic.\n\tMoreover, if $p_0$ is the transition density of isotropic unimodal L\\'evy process, then\n\t$\\RR^d \\ni y \\mapsto p_0(t; 0, y)$ is radial and non-increasing, thus $\\RR^d \\ni y \\mapsto p_j(t; 0, y)$ is radial and\n\tnon-increasing too. Indeed, by Proposition \\ref{prop:4}\\eqref{en:2:1},\n\t\\begin{align*}\n\t\tp_{j+1}(t; 0, y)\n\t\t&=\n\t\t\\int_0^t \\int_{\\RR^d} p_0(s; 0, z) p_j(t-s; cz, y) {\\: \\rm d} z {\\: \\rm d} s \\\\\n\t\t&\n\t\t=\n\t\t\\int_0^t \\int_{\\RR^d} p_0(s; 0, z) p_j\\big(t-s; 0, c^{j+1} (c^{-j-1}y - z) \\big) {\\: \\rm d} z {\\: \\rm d} s.\n\t\\end{align*}\n\tConsequently, the process $\\mathbf{X}$ is isotropic and unimodal if $\\mathbf{X}$ is such.\n\\end{remark}\n\n\\subsection{Representation of the transition density}\n\\label{sec:repr}\nWe are now ready to prove the key representation of the density. From this point we assume that the original process is\n$\\alpha$-stable for $\\alpha \\in (0, 2]$.\n\\begin{theorem}\n\t\\label{thm:rep-gen-st}\n\tSuppose that $\\mathbf{Y}$ is a strictly $\\alpha$-stable process in $\\RR^d$, $\\alpha \\in (0, 2]$, with a transition density\n\t$p_0$. Suppose that $\\mathbf{X}$ is obtained from $\\mathbf{Y}$ by partial resetting with factor $c \\in (0, 1)$. Then\n\tthe transition density of $\\mathbf{X}$ satisfies\n\t\\begin{align}\n\t\t\\label{eq:7}\n\t\tp_n(t; 0, y)\n\t\t= t^n \\int_0^1 p_0(t u; 0, y) P_n(u) {\\: \\rm d} u, \\quad \\text{for all } n \\in \\NN, y \\in \\RR^d, t > 0.\n\t\\end{align}\n\tMoreover, for all $x, y \\in \\RR^d$ and $t > 0$,\n\t\\begin{align}\n\t\t\\label{eq:rep-p-x}\n\t\tp(t; x, y)=e^{-t}p_0(t; 0, y-x)+e^{-t}\\sum_{j=1}^\\infty t^j \\int_0^1 p_0(tu;0,y-c^jx) P_j(u) {\\: \\rm d} u.\n\t\\end{align}\n\\end{theorem}\n\\begin{proof}\n\tFirst, we link $W_n$ with $p_n$ which are defined by \\eqref{eq:15}. Namely, we claim that\n\t\\begin{equation}\n\t\t\\label{eq:32}\n\t\tp_n(t; 0, y) = \\int_{m^n t}^t p_0(u; 0, y) W_n(t, u) {\\: \\rm d} u,\n\t\t\\quad \\text{for all } y \\in \\RR^d, t > 0.\n\t\\end{equation}\n\tThe proof is by induction with respect to $n \\in \\NN$. For $n = 1$, we have\n\t\\begin{align*}\n\t\tp_1(t; 0, y)\n\t\t&= \\int_0^t \\int_{\\RR^d} p_0(s; 0, z) p_0(t - s; m^{\\frac{1}{\\alpha}} z, y) {\\: \\rm d} z {\\: \\rm d} s \\\\\n\t\t&= \\int_0^t \\int_{\\RR^d}\n\t\tp_0(s; 0, z m^{-\\frac{1}{\\alpha}}) p_0(t - s; z, y) m^{-\\frac{d}{\\alpha}} {\\: \\rm d} z {\\: \\rm d} s.\n\t\\end{align*}\n\tSince $p_0$ is the density of $\\alpha$-stable process, it satisfies the scaling property\n\t(see \\cite[Theorems 14.2 and 14.7]{MR1739520}), hence\n\t\\begin{align*}\n\t\tp_1(t; 0, y)\n\t\t&=\n\t\t\\int_0^t \\int_{\\RR^d}\n\t\tp_0(m s; 0, z) p_0(t - s; z, y) {\\: \\rm d} z {\\: \\rm d} s \\\\\n\t\t&=\n\t\t\\int_0^t p_0(t + ms - s; 0, y) {\\: \\rm d} s \\\\\n\t\t&=\n\t\t\\int^t_{m t} p_0(s; 0, y) \\frac{1}{1-m} {\\: \\rm d} s.\n\t\\end{align*}\n\tNext, we write\n\t\\begin{align*}\n\t\tp_{n+1}(t; 0, y)\n\t\t&=\n\t\t\\int_0^t \\int_{\\RR^d} p_0(s; 0, z) p_n(t-s; 0, y- m^{\\frac{n+1}{\\alpha} z}) {\\: \\rm d} z {\\: \\rm d} s \\\\\n\t\t&=\n\t\t\\int_0^t \\int_{\\RR^d} p_0(s; 0, z) \\int_{m^n (t-s)}^{t-s} p_0(u; 0, y - m^{\\frac{n+1}{\\alpha}} z)\n\t\tW_n(t-s, u) {\\: \\rm d} u {\\: \\rm d} z {\\: \\rm d} s \\\\\n\t\t&=\n\t\t\\int_0^t \\int_{m^n (t-s)}^{t-s}\n\t\t\\int_{\\RR^d} p_0(s; 0, z) p_0(u; 0, y-m^{\\frac{n+1}{\\alpha}} z) {\\: \\rm d} z W_n(t-s, u)\n\t\t{\\: \\rm d} u {\\: \\rm d} s \\\\\n\t\t&=\n\t\t\\int_0^t \\int_{m^n (t-s)}^{t-s}\n\t\t\\int_{\\RR^d} m^{-d\\frac{n+1}{\\alpha}} p_0(s; 0, m^{-\\frac{n+1}{\\alpha}} z)\n\t\tp_0(u; 0, y - z) {\\: \\rm d} z W_n(t-s, u) {\\: \\rm d} u {\\: \\rm d} s \\\\\n\t\t&=\n\t\t\\int_0^t\n\t\t\\int_{m^n (t-s)}^{t-s} \\int_{\\RR^d} p_0(m^{n+1} s; 0, z)\n\t\tp_0(u; z, y) {\\: \\rm d} z W_n(t-s, u) {\\: \\rm d} u {\\: \\rm d} s\n\t\\end{align*}\n\twhere in the last equality we have used properties of $\\alpha$-stable density. Hence,\n\t\\begin{align*}\n\t\tp_{n+1}(t; 0, y)\n\t\t&=\n\t\t\\int_0^t\n\t\t\\int_{m^n (t-s)}^{t-s} p_0(u + m^{n+1} s; 0, y) W_n(t - s, u) {\\: \\rm d} u {\\: \\rm d} s \\\\\n\t\t&=\n\t\t\\int_0^t \\int^{t-s + m^{n+1}s}_{m^n (t-s)+m^{n+1}s} p_0(u; 0, y) W_n(t-s, u - m^{n+1}s) {\\: \\rm d} u {\\: \\rm d} s.\n\t\\end{align*}\n\tNow, we observe that\n\t\\begin{align*}\n\t\t&\n\t\t\\Big\\{ (s, u) \\in \\RR^2 :\n\t\t0 \\leq s \\leq t \\text{ and } m^n(t-s) + m^{n+1} s \\leq u \\leq t-s+m^{n+1}s \\Big\\} \\\\\n\t\t&\\qquad=\n\t\t\\bigg\\{(s, u) \\in \\RR^2 :\n\t\tm^{n+1} t \\leq u \\leq t, \\text{ and } \\frac{m^n t - u}{m^n - m^{n+1}} \\vee 0 \\leq s \\leq \\frac{t - u}{1- m^{n+1}}\n\t\t\\bigg\\}.\n\t\\end{align*}\n\tThus changing the order of integration we obtain\n\t\\begin{align*}\n\t\tp_{n+1}(t; 0, y)\n\t\t&=\n\t\t\\int_{m^{n+1} t}^t\n\t\t\\int_{\\frac{m^n t- u}{m^n - m^{n+1}} \\vee 0}^{\\frac{t-u}{1-m^{n+1}}}\n\t\tp_0(u; 0, y) W_n(t-s, u - m^{n+1}s) {\\: \\rm d} s {\\: \\rm d} u\n\t\\end{align*}\n\tas claimed.\n\t\n\tHaving proved \\eqref{eq:23}, by Proposition \\ref{prop:3} and \\eqref{eq:21}, we can write\n\t\\begin{align*}\n\t\tp_n(t; 0, y)\n\t\t&= \\int_{m^n t}^t p_0(u; 0, y) W_n(t, u) {\\: \\rm d} u \\\\\n\t\t&= \\int_{m^n t}^t p_0(u; 0, y) t^{n-1} P_n(u t^{-1}) {\\: \\rm d} u\n\t\t= t^n \\int_{m^n}^1 p_0(t u; 0, y) P_n(u) {\\: \\rm d} u.\n\t\\end{align*}\n\tNow, \\eqref{eq:rep-p-x} follows by Proposition \\ref{prop:5}, Proposition~\\ref{prop:4}\\eqref{en:2:1} and \\eqref{eq:7}.\n\\end{proof}\n\n\\begin{corollary}\n\t\\label{cor:rep-1}\n\tUnder the assumptions of Theorem \\ref{thm:rep-gen-st}, the transition density of the process $\\mathbf{X}$ satisfies\n\t\\begin{align}\n\t\t\\label{eq:rep-p-0.1}\n\t\tp(t;0,y)= \\int_0^\\infty p_0(u;0,y) \\: \\mu_t({\\rm d} u), \\quad\\text{for all } y \\in \\RR^d, t > 0\n\t\\end{align}\n\twhere $\\mu_t$ is the measure defined by the formula \\eqref{def:mu_t}.\n\\end{corollary}\n\n\\begin{corollary}\n\t\\label{cor:rep-2}\n\tUnder the assumptions of Theorem \\ref{thm:rep-gen-st}, the transition density of the process $\\mathbf{X}$ satisfies\n\t\\begin{align}\n\t\t\\label{eq:rep-p-0}\n\t\tp(t;0,y)= e^{-t}p_0(t;0,y)+e^{-t} \\int_0^1 p_0(tu;0,y) \\Phi(t,u) {\\: \\rm d} u, \\quad\\text{for all } y \\in \\RR^d, t > 0\n\t\\end{align}\n\twhere\n\t\\begin{align}\n\t\t\\label{def:Phi}\n\t\t\\Phi(t, u)= \\sum_{j = 1}^\\infty t^j P_j(u),\\qquad u \\in [0, 1],\\, t > 0.\n\t\\end{align}\n\\end{corollary}\n\n\\subsection{Ergodicity of $\\mathbf{X}$}\nIn this section we show that the process obtained by partial resetting of $\\alpha$-stable process, $\\alpha \\in (0, 2]$,\nis ergodic, see Theorem \\ref{thm:lim_p_t_infty}. In fact, we are going to prove that $p(t; x, y)$ converges uniformly in a\ncertain space-time region in $(t, y)$ as $t$ tends to infinity.\n\\begin{theorem}\n\t\\label{thm:lim_p_t_infty}\n\tSuppose that $\\mathbf{Y}$ is a strictly $\\alpha$-stable process in $\\RR^d$, $\\alpha \\in (0, 2]$, with a transition density.\n\tAssume that $\\mathbf{X}$ is obtained from $\\mathbf{Y}$ by partial resetting with factor $c \\in (0, 1)$.\n\tThen for each $\\delta > 0$,\n\t\\begin{align}\n\t\t\\label{eq:lim_p_t_infty-unif}\n\t\t\\lim_{t\\to+\\infty} \\sup_{y \\in \\RR^d}\\sup_{x \\in \\mathscr{X}(\\delta; t)}\n\t\t| p(t;x,y) - \\rho_{\\mathbf{Y}}(y)|=0\n\t\\end{align}\n\twhere $\\mathscr{X}(\\delta; t)$ is defined in \\eqref{eq:52} and $\\rho_{\\mathbf{Y}}$ is defined in \\eqref{representationrho}.\n\tIn particular, for all $x, y \\in \\RR^d$,\n\t\\begin{equation}\n\t\t\\label{eq:64}\n\t\t\\lim_{t\\to+\\infty} p(t;x,y)=\\rho_{\\mathbf{Y}}(y).\n\t\\end{equation}\n\\end{theorem}\n\\begin{proof}\n\tThe convergence \\eqref{eq:lim_p_t_infty-unif} is a consequence of Lemma \\ref{lem:unif_conv-2} applied to the family of\n\tfunctions\n\t\\[\n\t\t\\mathcal{F}=\\big\\{p_0(u; 0, y-x) : y\\in\\RR^d \\big\\}. \\qedhere\n\t\\]\n\\end{proof}\n\nBy Theorem \\ref{thm:lim_p_t_infty} together with Scheff\\'e's lemma, see e.g. \\cite{MR2728440}, we easily get the following\ncorollary.\n\\begin{corollary}\n\t\\label{cor:1}\n\tUnder the assumptions of Theorem \\ref{thm:lim_p_t_infty}, for all $x \\in \\RR^d$,\n\t\\begin{equation}\n\t\t\\label{eq:65}\n\t\t\\lim_{t\\to+\\infty} \\int_{\\RR^d}|p(t;x,y)-\\rho_{\\mathbf{Y}}(y)| {\\: \\rm d}y=0.\n\t\\end{equation}\n\tIn particular, the measure $\\rho_{\\mathbf{Y}}(y)$ is an invariant measure of the process $\\mathbf{X}$.\n\\end{corollary}\n\n\\subsection{The transition density of $\\mathbf{X}$}\n\\label{sec:3.3}\nWe study the regularity of the transition density of the process obtained by a partial resetting from $\\alpha$-stable process.\n\\begin{lemma}\n\t\\label{lem:p_reg}\n\tLet $\\mathbf{Y}$ be a strictly $\\alpha$-stable process in $\\RR^d$, $\\alpha \\in (0, 2]$, with a transition density\n\t$p_0$. Suppose that $\\mathbf{X}$ is obtained from $\\mathbf{Y}$ by partial resetting with factor $c \\in (0, 1)$.\n\tThen $p$ the transition density of $\\mathbf{X}$ belongs to $\\calC^{\\infty}\\big((0,\\infty)\\times\\RR^d\\times\\RR^d\\big)$.\n\tMoreover, for all $\\ell \\in \\NN_0$, $\\mathbf{a}, \\mathbf{b} \\in \\NN_0^d$, $x, y \\in \\RR^d$ and $t>0$, both functions\n\t\\[\n\t\tz \\mapsto \\partial_t^\\ell \\partial_{x}^{\\mathbf{a}}\\partial_{y}^\\mathbf{b} p(t; z,y),\n\t\t\\quad\n\t\t\\text{and}\n\t\t\\quad\n\t\tz \\mapsto \\partial_t^\\ell \\partial_{x}^\\mathbf{a} \\partial_{y}^\\mathbf{b} \\,p(t;x,z)\n\t\\]\n\tbelongs to $\\calC_0^\\infty(\\RR^d)$.\n\\end{lemma}\n\\begin{proof}\n\tFirst, using \\eqref{eq:rep-p-x} and the induction with respect to the order of the differentiation one can prove that\n\t$\\partial_t^\\ell \\partial_{x}^{\\mathbf{a}}\\partial_{y}^{\\mathbf{b}} p(t;x,y)$ equals to\n\t$\\partial_t^\\ell \\partial_{x}^{\\mathbf{a}}\\partial_{y}^{\\mathbf{b}} (e^{-t} p_0(t;0,y-x))$ plus a linear combination\n\tof expressions of the form\n\t\\begin{align}\n\t\t\\label{expr:aux}\n\t\te^{-t}\\sum_{j=1}^\\infty c_j t^{j-\\ell_1} \\int_0^1 u^{\\ell_2} f(tu,y-c^jx) P_j(u) \\: {\\rm d}u\n\t\\end{align}\n\twhere $\\ell_1 + \\ell_2 = \\ell$,\n\t\\[\n\t\tc_j= \\begin{cases}\n\t\t\t\\frac{j!}{(j-\\ell_1)!}(-c^j)^{|\\mathbf{a}|} & \\text{if } j \\geq \\ell_1, \\\\\n\t\t\t0 & \\text{if } j<\\ell_1,\n\t\t\\end{cases}\n\t\\]\n\tand\n\t\\[\n\t\tf(u,y)= \\big(\\partial_u^{\\ell_2} \\,\\partial_{y}^{\\mathbf{a}+\\mathbf{b}}\\, p_0\\big)(u;0,y).\n\t\\]\n\tThe inductive step will be verified once we justify the differentiation under both the series and the integral sign.\n\tBy Lemma~\\ref{lem:A3}\\eqref{en:3:2} we have\n\t\\begin{align}\n\t\t\\label{bound:aux}\n\t\t|\\partial_u f(u,y)| +|\\nabla_y f(u,y)|+|f(u,y)| \\leq C\\big(u+u^{-1}\\big)^\\gamma,\n\t\\end{align}\n\tfor certain $C,\\gamma > 0$. Recall that each $P_j$ has support bounded away from zero, see Proposition~\\ref{prop:1}.\n\tThus, due to \\eqref{bound:aux}, any partial sum of the series in \\eqref{expr:aux} may be differentiated as\n\tmany times as one needs.\n\tLet us consider $j \\geq \\ell_1+\\lceil \\gamma \\rceil+1$. By \\eqref{bound:aux} each difference quotient may be bounded\n\tterms of the form\n\t\\begin{align*}\n\t\t\\sum_{j=\\ell_1+\\lceil \\gamma \\rceil+1}^\\infty\n\t\t\\frac{j!}{(j-k)!} (2t)^{j-k}\n\t\t\\int_0^1 u^{\\ell_2-\\lceil\\gamma\\rceil}\\left(t^{\\lceil \\gamma \\rceil}+\n\t\tt^{-\\lceil\\gamma\\rceil}\\right) P_j(u) \\: {\\rm d}u\n\t\\end{align*}\n\twhere $k = \\ell_1$ or $k = \\ell_1+1$, which by Theorem~\\ref{thm:all-moments} is finite. Hence, to conclude it is enough to\n\tinvoke the dominated convergence theorem.\n\n\tLastly, the convergence to zero as $\\norm{x} + \\norm{y}$ gets large follows by reasoning similar to the proof of\n\tLemma~\\ref{lem:A3}\\eqref{en:3:2}, together with \\eqref{expr:aux} and \\eqref{bound:aux}.\n\\end{proof}\n\n\\begin{remark}\n\tThe Markov property of $\\mathbf{X}$ implies that\n\t\\[\n\t\tP_t f(x) = \\int_{\\RR^d} p(t; x, y) f(y) {\\: \\rm d} y, \\qquad x \\in \\RR^d\n\t\\]\n\tforms a semigroup on nonnegative bounded functions. In view of the regularity of $p$, the Chapmann--Kolmogorov equation\n\tholds true, that is\n\t\\begin{equation}\n\t\t\\label{eq:90}\n\t\tp(t + s; x, y) = \\int_{\\RR^d} p(t; x, z) p(s; z, y) {\\: \\rm d} z,\n\t\t\\qquad\\text{ for all } s, t > 0 \\text{ and } x, y \\in \\RR^d.\n\t\\end{equation}\n\\end{remark}\n\n\\subsection{Non-equilibrium stationary state of $\\mathbf{X}$}\n\\label{sec:3.4}\nIn this section our aim is to show that $\\mathbf{X}$ has non-equilibrium stationary state. To do so, we prove that\nthe infinitesimal generator of $\\mathbf{X}$ on $L^2(\\RR^d, \\rho_{\\mathbf{Y}}(y) {\\rm d} y)$ is \\emph{not} self-adjoint,\nsee Theorem \\ref{thm:NESS}. We also show that $\\rho_{\\mathbf{Y}}$ is harmonic with respect to $L^2$-adjoint operator\nto the infinitesimal generator of $\\mathbf{X}$, see Theorem \\ref{thm:H+F-P}.\n\nFor $f \\in \\calC_0^2(\\RR^d)$ we set\n\\[\n\t\\begin{aligned}\n\t\\mathscr{L} f(x)=\n\t\\sum_{j,k=1}^d a_{jk} \\partial_{x_j} \\partial_{x_k} f(x) &+ \\sprod{\\nabla f(x)}{\\gamma} \\\\\n\t&+\\int_{\\RR^d} \\big(f(x+z)-f(x)-\\ind{\\{|z|<1\\}} \\sprod{\\nabla f(x)}{z} \\big) \\: \\nu({\\rm d} z)\n\t\\end{aligned}\n\\]\nwhere $(A, \\gamma, \\nu)$ with $A=[a_{jk}]_{j,k=1}^d$ is the L{\\'e}vy triplet of $\\mathbf{Y}$, see Appendix \\ref{appendix:A}.\nFurthermore, we set\n\\[\n\t\\begin{aligned}\n\t\\mathscr{L}^*f(x)\n\t=\n\t\\sum_{j,k=1}^d a_{jk} \\partial_{x_j} \\partial_{x_k} f (x) &+ \\sprod{\\nabla f(x)}{\\gamma^*} \\\\\n\t&+\\int_{\\RR^d} \\big(f(x+z)-f(x)-\\ind{\\{|z|<1\\}} \\sprod{\\nabla f(x)}{z}\\big) \\: \\nu^*({\\rm d} z)\n\t\\end{aligned}\n\\]\nwhere $\\gamma^*=-\\gamma$ and $\\nu^*(B)=\\nu(-B)$, for a Borel set $B \\subset \\RR^d$. Accordingly, we define\n\\[\n\t\\mathscr{A} f(x) = \\mathscr{L} f(x) + f(cx) - f(x),\n\t\\qquad\\text{and}\\qquad\n\t\\mathscr{A}^* f(x) = \\mathscr{L}^* f(x) + \\frac{1}{c} f\\Big(\\frac{x}{c}\\Big) - f(x).\n\\]\nGiven $r \\in [1, \\infty)$, we let\n\\[\n\t\\mathcal{B}_r =\n\t\t\\begin{cases}\n\t\t\tL^r(\\RR^d) & \\text{if } r \\in [1, \\infty), \\\\\n\t\t\t\\calC_0(\\RR^d) & \\text{if } r = \\infty.\n\t\t\\end{cases}\n\\]\nIn view of Proposition \\ref{prop:9}, $\\mathbf{Y}$ generates a strongly continuous contractive semigroup on $\\mathcal{B}_r$.\nLet $\\mathscr{L}_r$ be the infinitesimal generator of this semigroup, and let $D(\\mathscr{L}_r)$ be the domain of $\\mathscr{L}_r$.\n\\begin{theorem}\n\t\\label{thm:1}\n\tSuppose that $\\mathbf{Y}$ is a strictly $\\alpha$-stable process in $\\RR^d$, $\\alpha \\in (0, 2]$, with a transition density.\n\tAssume that $\\mathbf{X}$ is obtained from $\\mathbf{Y}$ by partial resetting with factor $c \\in (0, 1)$. Let\n\t$r \\in [1, \\infty]$. Then the process $\\mathbf{X}$ generates strongly continuous semigroup of bounded operators on\n\t$\\mathcal{B}_r$. Its infinitesimal generator is\n\t\\begin{equation}\n\t\t\\label{eq:87}\n\t\t\\mathscr{A}_r f(x) = \\mathscr{L}_r f(x) + f(cx) - f(x), \\qquad f \\in D(\\mathscr{A}_r).\n\t\\end{equation}\n\tMoreover, $D(\\mathscr{A}_r) = D(\\mathscr{L}_r)$.\n\\end{theorem}\n\\begin{proof}\n\tFix $r \\in [1, \\infty]$. For $f \\in \\mathcal{B}_r$ and $t > 0$ we set\n\t\\[\n\t\tP_t f(x) = \\int_{\\RR^d} p(t; x, y) f(y) {\\: \\rm d} y, \\qquad\\text{ for } x \\in \\RR^d.\n\t\\]\n\tBy Lemma \\ref{lem:p_reg} and H\\\"older's inequality, the integral is well-defined. The proof that $(P_t : t > 0)$ is\n\ta strongly continuous semigroup of bounded operators on $\\mathcal{B}_r$, is essentially\n\tthe same as the proof that its infinitesimal generator is given by \\eqref{eq:87}, which is proved as follows:\n\tFirst, using \\eqref{eq:rep-p-x}, we write\n\t\\begin{align*}\n\t\t\\frac{P_t f(x) - f(x)}{t}\n\t\t&= e^{-t} \\frac{1}{t} \\int_{\\RR^d} p_0(t; x, y) \\big(f(y) - f(x)\\big) {\\: \\rm d} y \\\\\n\t\t&\\phantom{=}\n\t\t+ e^{-t} \\sum_{j = 1}^\\infty t^{j-1} \\int_{\\RR^d} \\int_0^1 p_0(tu; 0, y- c^j x) \\big(f(y) -f(x)\\big) P_j(u) {\\: \\rm d} u.\n\t\\end{align*}\n\tLet us observe that in the sum the terms for $j \\geq 2$ tends to zero in $\\mathcal{B}_r$. Indeed, we have\n\t\\begin{align*}\n\t\t&\n\t\t\\bigg\\| \\int_{\\RR^d} \\int_0^1 p_0(tu; 0, y - c^j x) \\big(f(y) - f(x) \\big) P_j(u) {\\: \\rm d} u {\\: \\rm d} y\n\t\t\\bigg\\|_{\\mathcal{B}_r(x)} \\\\\n\t\t&\\qquad=\n\t\t\\bigg\\| \\int_{\\RR^d} \\int_0^1 p_0(tu; 0, y) \\big(f(y+c^j x) - f(x) \\big) P_j(u) {\\: \\rm d} u {\\: \\rm d} y\n\t\t\\bigg\\|_{\\mathcal{B}_r(x)} \\\\\n\t\t&\\qquad\\leq\n\t\t\\int_{\\RR^d} \\int_0^1 p_0(tu; 0, y) \\big\\|f(y+c^j x) - f(x) \\big\\|_{\\mathcal{B}_r(x)} {\\: \\rm d} u {\\: \\rm d} y \\\\\n\t\t&\\qquad\\leq\n\t\t\\frac{1 + c^{-d j/r}}{j!} \\|f\\|_{\\mathcal{B}_r}\n\t\\end{align*}\n\twhere in the last inequality we have used Corollary \\ref{cor:A0}. Hence,\n\t\\begin{align*}\n\t\t\\bigg\\|\n\t\t\\sum_{j = 2}^\\infty t^{j-1} \\int_{\\RR^d} \\int_0^1 p_0(tu; 0, y- c^j x) \\big(f(y) -f(x)\\big) P_j(u)\n\t\t{\\: \\rm d} u\n\t\t{\\: \\rm d} y\n\t\t\\bigg\\|_{\\mathcal{B}_r(x)}\n\t\t\\leq\n\t\t\\|f\\|_{\\mathcal{B}_r} t\n\t\t\\sum_{j = 0}^\\infty\n\t\t\\frac{1 + c^{-d (j+2)/r}}{(j+2)!} t^j.\n\t\\end{align*}\n\tFor $j = 1$, we write\n\t\\begin{align*}\n\t\t&\\bigg\\|\n\t\t\\int_{\\RR^d} \\int_0^1 p_0(tu; 0, y) f(y + c x) P_1(u) {\\: \\rm d} u {\\: \\rm d} y\n\t\t-f(cx)\n\t\t\\bigg\\|_{\\mathcal{B}_r(x)} \\\\\n\t\t&\\qquad\\leq\n\t\t\\int_0^1 \\bigg\\| \\int_{\\RR^d} p_0(tu; 0, y) \\big( f(y + c x) - f(cx)\\big) {\\: \\rm d} y \\bigg\\|_{\\mathcal{B}_r(x)}\n\t\tP_1(u) {\\: \\rm d} u,\n\t\\end{align*}\n\tthus by the dominated convergence together with the strong continuity on $\\mathcal{B}_r$, we conclude that\n\t\\[\n\t\t\\lim_{t \\to 0^+} \\int_{\\RR^d} \\int_0^1 p_0(tu; 0, y) f(y + c x) P_1(u) {\\: \\rm d} u {\\: \\rm d} y\n\t\t=\n\t\tf(cx),\n\t\\]\n\tand the theorem follows.\n\\end{proof}\n\n\\begin{corollary}\n\t\\label{cor:2}\n\tUnder the assumptions of Theorem \\ref{thm:1}, for every $r \\in [1, \\infty]$, if $f \\in \\calC_0^2(\\RR^d)$ is such that\n\t$\\partial_x^{\\mathbf{a}} f \\in \\mathcal{B}_r$ for each $\\mathbf{a} \\in \\NN_0^d$, $\\abs{\\mathbf{a}} \\leq 2$, then\n\t$f \\in D(\\mathscr{A}_r)$ and\n\t\\[\n\t\t\\mathscr{A}_r f = \\mathscr{A} f.\n\t\\]\n\\end{corollary}\nLet $\\mathscr{A}^*_2$ denote the $L^2(\\RR^d)$-adjoint operator to $\\mathscr{A}_2$. Recall that $g \\in D(\\mathscr{A}_2^*)$ if\nand only if there is $h \\in L^2(\\RR^d)$ such that\n\\[\n\t\\sprod{\\mathscr{A}_2 f}{g} = \\sprod{f}{h} \\qquad\\text{for all } f \\in D(\\mathscr{A}_2).\n\\]\nThen $\\mathscr{A}_2^* g = h$. We have the following lemma.\n\\begin{lemma}\n\t\\label{lem:3}\n\tSuppose that $\\mathbf{Y}$ is a strictly $\\alpha$-stable process in $\\RR^d$, $\\alpha \\in (0, 2]$, with a transition density.\n\tAssume that $\\mathbf{X}$ is obtained from $\\mathbf{Y}$ by partial resetting with factor $c \\in (0, 1)$. If\n\t$g \\in \\calC_0^2(\\RR^d)$ is such that $\\partial^{\\mathbf{a}} g \\in L^2(\\RR^d)$ for each $\\mathbf{a} \\in \\NN_0^d$,\n\t$\\abs{\\mathbf{a}} \\leq 2$, then $g \\in D(\\mathscr{A}_2^*)$ and $\\mathscr{A}_2^* g = \\mathscr{A}^* g$.\n\\end{lemma}\n\\begin{proof}\n\tLet $f \\in D(\\mathscr{A}_2)$. By Proposition \\ref{prop:9}, we have\n\t\\begin{align*}\n\t\t\\sprod{\\mathscr{L}_2 f}{g}\n\t\t&=\n\t\t\\lim_{t\\to 0^+}\n\t\t\\int_{\\RR^d}\n\t\t\\frac{\\EE\\big[f(Y_t + x)\\big]-f(x)}{t} g(x) \\: {\\rm d} x\\\\\n\t\t&=\n\t\t\\lim_{t\\to 0^+} \\int_{\\RR^d} f(x) \\frac{\\EE\\big[g(-Y_t + x)\\big] -g(x)}{t} \\: {\\rm d} x\n\t\t=\n\t\t\\sprod{f}{\\mathscr{L}^*g}.\n\t\\end{align*}\n\tHence, by Theorem \\ref{thm:1},\n\t\\begin{align*}\n\t\t\\sprod{\\mathscr{A}_2 f}{g}\n\t\t&= \\sprod{\\mathscr{L}_2 f}{g}+ \\int_{\\RR^d} \\big(f(cx) - f(x)\\big) g(x) \\: {\\rm d}x \\\\\n\t\t&= \\sprod{f}{\\mathscr{L}^*g}+ \\int_{\\RR^d} f(x) \\bigg(\\frac{1}{c} g(cx)-g(x)\\bigg) \\: {\\rm d}x\n\t\t= \\sprod{f}{\\mathscr{A}^*g}.\n\t\\end{align*}\n\tSince $\\mathscr{A}^*g \\in L^2(\\RR^d)$, we conclude that $\\mathscr{A}_2^* g = \\mathscr{A}^* g$.\n\\end{proof}\n\n\\begin{theorem}\n\t\\label{thm:H+F-P}\n\tSuppose that $\\mathbf{Y}$ is a strictly $\\alpha$-stable process in $\\RR^d$, $\\alpha \\in (0, 2]$, with a transition density.\n\tAssume that $\\mathbf{X}$ is obtained from $\\mathbf{Y}$ by partial resetting with factor $c \\in (0, 1)$. Then\n\t$\\rho_{\\mathbf{Y}}$ defined in \\eqref{representationrho} belongs to $D(\\mathscr{A}_2^*)$ and\n\t\\begin{equation}\n\t\t\\label{eq:69}\n\t\t\\mathscr{A}_2^* \\rho_{\\mathbf{Y}} = \\mathscr{A}^* \\rho_{\\mathbf{Y}}= 0.\n\t\\end{equation}\n\tMoreover,\n\t\\[\n\t\t\\partial_t p = \\mathscr{A}_x p =\\mathscr{A}_y^* p\n\t\\]\n\\end{theorem}\n\\begin{proof}\n\tIn view of Proposition \\ref{prop:6} and Lemma \\ref{lem:p_reg}, $\\rho_{\\mathbf{Y}}$ and $p$ are regular, respectively.\n\tLet $(P_t : t > 0)$ be the semigroup on $\\calC_0(\\RR^d)$ generated by $\\mathbf{X}$. By Corollary \\ref{cor:2},\n\tfor $f \\in \\calC_c^\\infty(\\RR^d)$,\n\t\\[\n\t\t\\partial_t P_t f(x) = P_t \\mathscr{A} f(x).\n\t\\]\n\tHence, by Lemma \\ref{lem:p_reg},\n\t\\begin{align*}\n\t\t\\int_{\\RR^d} f(y) \\partial_t p(t; x, y) {\\: \\rm d} y\n\t\t&=\n\t\t\\partial_t P_t f(x)\n\t\t=\n\t\tP_t \\mathscr{A} f(x) \\\\\n\t\t&= \\int_{\\RR^d} \\mathscr{A} f(y) p(t; x, y) {\\: \\rm d} y\n\t\t= \\int_{\\RR^d} f(y) \\mathscr{A}^*_y p(t; x, y) {\\: \\rm d} y\n\t\\end{align*}\n\tproving $\\partial_t p = \\mathscr{A}_y^* p$. To show that $\\partial_t p = \\mathscr{A}_x p$, we observe that for\n\t$f \\in \\calC_0^\\infty(\\RR^d)$,\n\t\\[\n\t\t\\partial_t P_t f(x) = \\mathscr{A} P_t f(x)\n\t\\]\n\tprovided that $P_t f \\in \\calC_0^\\infty(\\RR^d)$. Therefore, in view of the Chapmann--Kolmogorov equation \\eqref{eq:90},\n\tit is enough to take $f(x) = p(s; x, y)$, since the required regularity is a consequence of Lemma \\ref{lem:p_reg}.\n\n\tTo prove \\eqref{eq:69}, by Corollary \\ref{cor:1}, for $f \\in \\calC_c^\\infty(\\RR^d)$ we have\n\t\\begin{align*}\n\t\t0\n\t\t=\n\t\t\\int_{\\RR^d} \\frac{P_t f(x) - f(x)}{t} \\rho_{\\mathbf{Y}}(x) {\\: \\rm d} x,\n\t\\end{align*}\n\tthus,\n\t\\[\n\t\t0=\\sprod{\\mathscr{A} f}{ \\rho_{\\mathbf{Y}}}\n\t\t=\n\t\t\\sprod{f}{\\mathscr{A}^* \\rho_{\\mathbf{Y}}}\n\t\t=\n\t\t\\sprod{f}{\\mathscr{A}_2^* \\rho_{\\mathbf{Y}}}\n\t\\]\n\twhere the last equation follows by Lemma \\ref{lem:3}, which completes the proof.\n\\end{proof}\nNotice that $\\mathbf{X}$ generates a strongly continuous semigroup of contractive operators on\n$L^2(\\RR^d, \\rho_{\\mathbf{Y}}(y) {\\rm d} y)$. Let us denote by $\\mathscr{A}_\\rho$ its infinitesimal generator.\nSince $\\rho_{\\mathbf{Y}}(y) {\\rm d} y$ is a probabilistic measure, for\n$f \\in \\calC_0^\\infty(\\RR^d) \\subset D(\\mathscr{A}_\\rho)$, we have $\\mathscr{A}_\\rho f = \\mathscr{A}_\\infty f = \\mathscr{A} f$.\nLet $\\mathscr{A}_\\rho^*$ denote the operator adjoint to $\\mathscr{A}_\\rho$ on $L^2(\\RR^d, \\rho_{\\mathbf{Y}}(y) {\\rm d} y)$. If\n$g \\in \\calC_c^\\infty(\\RR^d)$ belongs to the domain of $\\mathscr{A}^*_\\rho$, then\n\\begin{align*}\n\t\\int_{\\RR^d} \\mathscr{A}_\\rho^* g (y) f(y) \\: \\rho_{\\mathbf{Y}}(y) {\\rm d} y\n\t&=\n\t\\int_{\\RR^d} g(y) \\mathscr{A}_\\rho f(y) \\: \\rho_{\\mathbf{Y}} (y) {\\rm d} y \\\\\n\t&=\n\t\\int_{\\RR^d} \\rho_{\\mathbf{Y}} (y) g(y) \\mathscr{A} f(y) {\\rm d} y \\\\\n\t&=\n\t\\int_{\\RR^d} \\mathscr{A}^* \\big(\\rho_{\\mathbf{Y}} g\\big)(y) f(y) {\\rm d} y\n\\end{align*}\nfor all $f \\in \\calC_c^\\infty(\\RR^d)$. Hence,\n\\begin{equation}\n\t\\label{eq:91}\n\t\\rho_{\\mathbf{Y}} \\: \\mathscr{A}_\\rho^* g = \\mathscr{A}^* \\big(\\rho_{\\mathbf{Y}} \\: g\\big).\n\\end{equation}\n\\begin{theorem}\n\t\\label{thm:NESS}\n\tSuppose that $\\mathbf{Y}$ is a strictly $\\alpha$-stable process in $\\RR^d$, $\\alpha \\in (0, 2]$, with a transition density.\n\tAssume that $\\mathbf{X}$ is obtained from $\\mathbf{Y}$ by partial resetting with factor $c \\in (0, 1)$.\n\tThere exists $g \\in D(\\mathscr{A}_\\rho)$ such that if $g \\in D(\\mathscr{A}_\\rho^*)$ then\n\t$\\mathscr{A}_\\rho g \\neq \\mathscr{A}_\\rho^* g$.\n\\end{theorem}\n\\begin{proof}\n\tWe are going to construct $g \\in \\calC_c^\\infty(\\RR^d)$.\n\tSince for $g \\in \\calC_c^\\infty(\\RR^d)$ both $\\mathscr{A}_\\rho g$ and $\\mathscr{A}_\\rho^* g$ are at least continuous,\n\tif suffices to show that $\\mathscr{A}_\\rho g(x) \\neq \\mathscr{A}_\\rho^* g(x)$ for some $x\\in \\RR^d$ such that\n\t$\\rho_{\\mathbf{Y}}(x)>0$.\n\tLet $B_r(x)$ denote the open Euclidean ball of radius $r > 0$ and centered at $x \\in \\RR^d$. Fix\n\t$x_0 \\in \\RR^d \\setminus\\{0\\}$ such that $\\rho_{\\mathbf{Y}}(x_0) > 0$. Let us consider a function\n\t$g_r \\in \\calC_c^\\infty(\\RR^d)$ such that\n\t\\[\n\t\t\\ind{B_{r/2}(c x_0)} \\leq g_r \\leq \\ind{B_r(c x_0)}.\n\t\\]\n\tIf $(1-c) \\norm{x_0} > r > 0$, then\n\t\\begin{align*}\n\t\t\\mathscr{A}_{\\rho} g_r(x_0)\n\t\t&=\n\t\t\\mathscr{A} g_r(x_0) \\\\\n\t\t&=\\mathscr{L} g_r(x_0)+g_r(cx_0)-g_r(x_0) \\\\\n\t\t&=\\int_{\\RR^d} g_r(x_0+z) \\: \\nu({\\rm d}z) + 1.\n\t\\end{align*}\n\tMoreover, by \\eqref{eq:91}\n\t\\begin{align*}\n\t\t\\mathscr{A}_\\rho^* g_r(x_0)\n\t\t&=\\frac1{\\rho_{\\mathbf{Y}}(x_0)} \\mathscr{A}^*(\\rho_{\\mathbf{Y}} g_r)(x_0)\\\\\n\t\t&=\\frac1{\\rho_{\\mathbf{Y}}(x_0)}\\bigg( \\mathscr{L}^*(\\rho_{\\mathbf{Y}} g_r)(x_0)\n\t\t+\\frac1{c}(\\rho_{\\mathbf{Y}} g_r)\\Big(\\frac{x_0}{c}\\Big)-(\\rho_{\\mathbf{Y}} g_r)(x_0)\\bigg)\\\\\n\t\t&=\\frac1{\\rho_{\\mathbf{Y}}(x_0)} \\int_{\\RR^d} \\rho_{\\mathbf{Y}}(x_0+z)g_r(x_0+z) \\: \\nu^*({\\rm d}z).\n\t\\end{align*}\n\tSince $\\nu$ and $\\nu^*$ both are measures without atoms, see \\eqref{eq:nu-str_st}, we obtain\n\t\\[\n\t\t\\lim_{r \\to 0^+}\n\t\t\\mathscr{A}_\\rho g_r(x_0)\n\t\t=\n\t\t1, \\qquad \\text{ and }\\qquad \\lim_{r\\to 0^+} \\mathscr{A}_\\rho^* g_r(x_0)=0.\n\t\\]\n\tTherefore, there is $r > 0$ such that $\\mathscr{A}_\\rho g_r(x_0) \\neq \\mathscr{A}_\\rho^* g_r(x_0)$.\n\\end{proof}\n\n\\section{Asymptotic and estimates of transition density}\\label{sec:4}\nIn this section we study the asymptotic behavior of the transition densities processes obtained by partial resetting from\n$\\alpha$-stable processes such as: isotropic $\\alpha$-stable processes (see Section \\ref{sec:stable}), $\\alpha$-stable\nsubordinators (see Section \\ref{sec:sub}) and $d$-cylindrical $\\alpha$-stable processes (see Section \\ref{sec:cyl}).\n\n\\subsection{Isotropic $\\alpha$-stable process with $\\alpha\\in(0,2)$ and resetting}\n\\label{sec:stable}\nLet $\\nu$ denote the L{\\'e}vy measure of an isotropic $\\alpha$-stable process with $\\alpha\\in (0,2)$. Namely,\n\\begin{align}\n\t\\label{eq:lm}\n\t\\nu(y)=\\frac{2^{\\alpha}\\Gamma((d+\\alpha)/2)}{\\pi^{d/2}|\\Gamma(-\\alpha/2)|} |y|^{-d-\\alpha},\n\\end{align}\nsee Example~\\ref{ex:names:2}. In view of the construction of the resetting, the transition density of the process $\\mathbf{X}$\ndepends on the parameter $c \\in (0, 1)$. Since in the following theorem we also study the asymptotic behavior of $p(t; x, y)$\nuniformly with respect to the parameter $c$, we write $p^{(c)}(t; x, y)$ instead of $p(t; x, y)$, to indicate\nthe dependence on $m = c^\\alpha$.\n\\begin{theorem}\n\t\\label{thm:ius}\n\tSuppose that $\\mathbf{Y}$ is an isotropic $\\alpha$-stable process in $\\RR^d$, $\\alpha \\in (0, 2)$, with a transition\n\tdensity $p_0$. Assume that $\\mathbf{X}^{(c)}$ is obtained from $\\mathbf{Y}$ by partial resetting with factor $c\\in(0,1)$.\n\tThen for each $\\kappa_1, \\kappa_2 \\in (0, 1)$, the transition density $p^{(c)}$ of $\\mathbf{X}^{(c)}$ satisfies\n\t\\begin{align}\n\t\t\\label{eq:ius-1}\n\t\t\\lim_{\\atop{|y| \\to +\\infty}{t \\to +\\infty}} \\,\n\t\t\\sup_{\\substack{c\\in(0,\\kappa_1] \\\\ |x| \\leq \\kappa_2 |y|}}\n\t\t\\left| (1-m)\\frac{p^{(c)}(t; x, y)}{\\nu(y)}-1 \\right|= 0.\n\t\\end{align}\n\tFurthermore, for any fixed $K>0$, we have\n\t\\[\n\t\t\\lim_{\\substack{ |y| \\to +\\infty \\\\ t \\to+\\infty}}\\,\n\t\t\\sup_{\\substack{c\\in (0,1)\\\\ |x|\\leq K |y|}}\n\t\t\\left| \\frac{(1-m)t}{1 -e^{-(1-m)t}}\\frac{p^{(c)}(t; x, t^{1/\\alpha} y)}{t\\nu(t^{1/\\alpha}y)} -1 \\right| = 0,\n\t\\]\n\tand\n\t\\[\n\t\t\\lim_{\\substack{|y| \\to +\\infty \\\\ t \\to+\\infty} }\n\t\t\\sup_{\\substack{c\\in (0,1)\\\\ |x|\\leq K }}\n\t\t\\left| \\frac{(1-m)t}{1 -e^{-(1-m)t}}\\frac{p^{(c)}(t; x, (1-m)^{-1/\\alpha}y)}{t\\nu((1-m)^{-1/\\alpha}y)} - 1 \\right| = 0.\n\t\\]\n\\end{theorem}\n\\begin{proof}\n\tIt is well known that (see \\cite{BlumenthalGetoor}),\n\t\\begin{align}\n\t\t\\label{approx:is}\n\t\tp_0(s;0,w)\\approx \\min\\bigg\\{ s^{-d/\\alpha}, \\frac{s}{|w|^{d+\\alpha}}\\bigg\\}\n\t\\end{align}\n\tuniformly with respect to $s > 0$ and $w \\in \\RR^d$. In particular, there is $C > 0$ such that for all $s > 0$ and\n\t$w \\in \\RR^d$,\n\t\\begin{align}\n\t\t\\label{ineq:2}\n\t\tp_0(s;0,w)\\leq C s \\nu(w).\n\t\\end{align}\n\tWe are going to prove all three statements simultaneously. We set\n\t\\[\n\t\t\\tilde{y}=y\\, \\qquad\\qquad \\tilde{y}=t^{1/\\alpha}y\\,, \\qquad\\qquad \\tilde{y}=(1-m)^{-1/\\alpha}y,\n\t\\]\n\trespectively. Fix $\\kappa_1, \\kappa_2 \\in (0,1)$ and $K>0$. By \\eqref{eq:rep-p-x}, we get\n\t\\begin{equation}\n\t\t\\label{eq:70}\n\t\tp^{(c)}(t; x, \\tilde{y})\n\t\t= e^{-t} p_0(t; 0, \\tilde{y}-x) +\n\t\te^{-t} \\sum_{j = 1}^\\infty t^j \\int_0^1 p_0(t u; 0, \\tilde{y}-c^j x) P_j(u) {\\: \\rm d} u.\n\t\\end{equation}\n\tWe emphasize that the splines $(P_j)$ depend on $c$. If $|y| \\geq K/\\kappa_2$ and $t \\geq (K/\\kappa_2)^{\\alpha}$, then\n\t$|x| \\leq \\kappa_2 |\\tilde{y}|$. Hence, for $j \\geq \\NN_0$,\n\t\\begin{align}\n\t\t\\label{ineq:1}\n\t\t\\frac{\\nu(\\tilde{y}-c^jx)}{\\nu(\\tilde{y})} \\leq (1-\\kappa_2)^{-d-\\alpha}.\n\t\\end{align}\n\tLet us observe that for $u > 0$,\n\t\\[\n\t\t\\frac{u}{1-e^{-u}} = u + \\frac{u}{e^u - 1} \\leq u + 1.\n\t\\]\n\tTherefore, by \\eqref{ineq:2} and \\eqref{ineq:1}, we get\n\t\\[\n\t\t\\frac{(1-m)t}{1 -e^{-(1-m)t}}\n\t\te^{-t}\\frac{p_0(t;0,\\tilde{y}-x)}{t\\nu(\\tilde{y})}\n\t\t\\leq\n\t\tC (1 - \\kappa_2)^{-d-\\alpha} (1+t) e^{-t},\n\t\\]\n\tand so, the first term in \\eqref{eq:70} uniformly tends to zero. Next, by \\eqref{ineq:2}, \\eqref{ineq:1} and\n\tTheorem \\ref{thm:all-moments} we obtain\n\t\\begin{align}\n\t\t\\nonumber\n\t\t&\\frac{(1-m) t}{1 -e^{-(1-m)t}} e^{-t} t^j\n\t\t\\int_0^1 \\bigg( \\frac{p_0(t u; 0, \\tilde{y}-c^j x)}{tu\\nu(\\tilde{y})}+ 1 \\bigg) u P_j(u) {\\: \\rm d} u \\\\\n\t\t\\nonumber\n\t\t&\\qquad\\qquad\\leq\n\t\t(1+t) e^{-t} t^j \\int_0^1 \\Bigg(\\frac{\\nu(\\tilde{y} - c^j x)}{\\nu(\\tilde{y})} + 1\\bigg) u P_j(u) {\\: \\rm d} u \\\\\n\t\t\\label{ineq:b-2}\n\t\t&\\qquad\\qquad\\leq\n\t\t\\big(C (1 - \\kappa_2)^{-d-\\alpha} + 1 \\big) (1+t) e^{-t} \\frac{t^j}{j!}.\n\t\\end{align}\n\tNext, let us recall that by \\cite{BlumenthalGetoor}\n\t\\[\n\t\t\\lim_{s |w|^{-\\alpha} \\to 0^+} \\frac{p_0(s; 0, w)}{s \\nu(w)} = 1.\n\t\\]\n\tFor $j \\in \\NN$, we can write\n\t\\begin{align*}\n\t\t\\bigg|\\frac{p_0(s; 0,\\tilde{y}-c^j x)}{s \\nu(\\tilde{y})} - 1 \\bigg|\n\t\t\\leq\n\t\t\\bigg|\\frac{p_0(s; 0, \\tilde{y}-c^j x)}{s \\nu(\\tilde{y}-c^j x)} - 1 \\bigg|\\cdot\n\t\t\\frac{\\nu(\\tilde{y}-c^j x)}{\\nu(\\tilde{y})}\n\t\t+\\bigg|\\frac{\\nu(\\tilde{y}-c^j x)}{\\nu(\\tilde{y})} -1\\bigg|.\n\t\\end{align*}\n\tSince $|x|\\leq \\kappa_2|\\tilde{y}|$, we get\n\t\\[\n\t\t\\frac{s}{|\\tilde{y}-c^jx|^{\\alpha}} \\leq \\frac{s |\\tilde{y}|^{-\\alpha}}{(1-\\kappa_2)^{\\alpha}}.\n\t\\]\n\tHence,\n\t\\begin{itemize}\n\t\t\\item if $|x|\\leq \\kappa_2|y|$, then\n\t\t\\[\n\t\t\t\\frac{\\nu(\\tilde{y})}{(1+\\kappa_1^j\\kappa_2)^{d+\\alpha}}\n\t\t\t\\leq \\nu(\\tilde{y}-c^j x) \\leq \\frac{\\nu(\\tilde{y})}{(1-\\kappa_1^j\\kappa_2)^{d+\\alpha}}\n\t\t\t\\quad\\text{ for all } c \\in (0,\\kappa_1];\n\t\t\\]\n\t\t\\item if $|x|\\leq K |y|$, then\n\t\t\\[\n\t\t\t\\frac{\\nu(\\tilde{y})}{(1+Kt^{-1/\\alpha})^{d+\\alpha}}\n\t\t\t\\leq \\nu(\\tilde{y}-c^jx) \\\\\n\t\t\t\\leq \\frac{\\nu(\\tilde{y})}{(1-Kt^{-1/\\alpha})^{d+\\alpha}}\n\t\t\t\\quad \\text{ for all }t > K^\\alpha;\n\t\t\\]\n\t\t\\item if $|x|\\leq K$, then\n\t\t\\[\n\t\t\t\\frac{\\nu(\\tilde{y})}{(1+K/|y|)^{d+\\alpha}} \\leq \\nu(\\tilde{y}-c^jx) \\leq \\frac{\\nu(\\tilde{y})}{(1-K/|y|)^{d+\\alpha}}\n\t\t\t\\quad \\text{ for all } |y|>K.\n\t\t\\]\n\t\\end{itemize}\n\tTherefore, for a given $\\epsilon>0$ there are $\\delta > 0$ and $n_0\\in\\NN$ depending only on $d$, $\\alpha$, $\\epsilon$,\n\t$\\kappa_1$, $\\kappa_2$ and $K$, such that\n\t\\begin{align}\n\t\t\\label{ineq:3}\n\t\t\\bigg|\\frac{p_0(s; 0, \\tilde{y}-c^j x)}{s \\nu(\\tilde{y})} - 1 \\bigg| \\leq \\epsilon\n\t\\end{align}\n\t\\begin{itemize}\n\t\t\\item for all $j \\geq n_0$, and $c \\in (0, \\kappa_1]$ provided that $s |\\tilde{y}|^{-\\alpha} \\leq \\delta$;\n\t\t\\item for all $j \\in \\NN$, $c \\in (0, 1)$ and $t \\geq \\delta^{-1}$ provided that $\\norm{x} \\leq K \\norm{y}$;\n\t\t\\item for all $j \\in \\NN$, $c \\in (0, 1)$ and $\\norm{y} \\geq \\delta^{-1}$ provided that $\\norm{x} \\leq K$.\n\t\\end{itemize}\n\tIn all three cases, we further investigate (see \\eqref{def:Phi} for the definition of $\\Phi$)\n\t\\begin{equation}\n\t\t\\label{eq:26}\n\t\t\\begin{aligned}\n\t\t&\n\t\t\\frac{(1-m)t}{1 -e^{-(1-m)t}} e^{-t}\n\t\t\\sum_{j = 1}^\\infty t^j \\int_0^1 \\frac{p_0(t u; 0, \\tilde{y}-c^j x)}{t\\nu(\\tilde{y})} P_j(u) {\\: \\rm d} u\n\t\t- \\frac{(1-m)t}{1 -e^{-(1-m)t}} e^{-t}\\int_0^1 u \\Phi(t, u) {\\: \\rm d} u\\\\\n\t\t&\\qquad\\qquad\n\t\t=\\frac{(1-m)t}{1 -e^{-(1-m)t}} e^{-t} \\sum_{j =1}^{n_0} t^j \\int_0^1\n\t\t\\bigg( \\frac{p_0(t u; 0, \\tilde{y}-c^j x)}{tu\\nu(\\tilde{y})}- 1 \\bigg) u P_j(u) {\\: \\rm d} u \\\\\n\t\t&\\qquad\\qquad\\phantom{=}\n\t\t+ \\frac{(1-m)t}{1 -e^{-(1-m)t}} e^{-t} \\sum_{j =n_0+1}^{\\infty} t^j\n\t\t\\int_{\\frac{\\delta |\\tilde{y}|^{\\alpha}}{t} \\land 1}^1\n\t\t\\bigg( \\frac{p_0(t u; 0, \\tilde{y}-c^j x)}{tu\\nu(\\tilde{y})}- 1 \\bigg) u P_j(u) {\\: \\rm d} u\\\\\n\t\t&\\qquad\\qquad\\phantom{=}\n\t\t+\\frac{(1-m)t}{1 -e^{-(1-m)t}}e^{-t} \\sum_{j =n_0+1}^{\\infty} t^j \\int_0^{\\frac{\\delta |\\tilde{y}|^{\\alpha}}{t} \\land 1}\n\t\t\\bigg( \\frac{p_0(t u; 0, \\tilde{y}-c^j x)}{tu\\nu(\\tilde{y})}- 1 \\bigg) u P_j(u) {\\: \\rm d} u.\n\t\t\\end{aligned}\n\t\\end{equation}\n\tThe first term is small due to \\eqref{ineq:b-2}. Furthermore, in all considered cases we can assure that\n\t$|y| \\geq \\delta$ which together with \\eqref{ineq:3} and \\eqref{eq:1-moment} (see \\eqref{def:mu_t} for the definition\n\tof $\\mu_t$) allows us to bound the absolute value of third term by\n\t\\begin{align*}\n\t\t&\\frac{(1-m)t}{1 -e^{-(1-m)t}}e^{-t}\\sum_{j =n_0+1}^{\\infty}\n\t\t\\int_0^{\\frac{\\delta |\\tilde{y}|^{\\alpha}}{t} \\land 1} t^j\n\t\t\\, \\bigg| \\frac{p_0(t u; 0, \\tilde{y}-c^j x)}{tu\\nu(\\tilde{y})}- 1 \\bigg| \\, u P_j(u) {\\: \\rm d} u \\\\\n\t\t&\\qquad\\qquad\n\t\t\\leq\n\t\t\\frac{(1-m) \\epsilon}{1-e^{-(1-m)t}} \\int_0^\\infty u \\: \\mu_t({\\rm d} u) = \\epsilon.\n\t\\end{align*}\n\tNext, the middle term in \\eqref{eq:26} equals zero in the case when $\\tilde{y}=t^{1/\\alpha}y$. If $\\tilde{y}=y$ or\n\t$\\tilde{y}=(1-m)^{-1/\\alpha}y$, by \\eqref{ineq:2}, \\eqref{ineq:1} and \\eqref{ineq:2-moment} we obtain\n\t\\begin{align*}\n\t\t&\\frac{(1-m)t}{1 -e^{-(1-m)t}}e^{-t} \\sum_{j =1}^\\infty t^j \\int_{\\frac{\\delta |\\tilde{y}|^{\\alpha}}{t} \\land 1}^1\n\t\t\\bigg( \\frac{p_0(t u; 0, \\tilde{y}-c^j x)}{tu\\nu(\\tilde{y})}+ 1 \\bigg) u P_j(u) {\\: \\rm d} u \\\\\n\t\t&\\qquad\\qquad \\leq \\frac{1-m}{1 -e^{-(1-m)t}}\n\t\t\\big(C(1-\\kappa_2)^{-d-\\alpha}+1\\big) \\int_{\\delta |\\tilde{y}|^{\\alpha} \\land t}^t u \\: \\mu_t({\\rm d} u) \\\\\n\t\t&\\qquad\\qquad\\leq \\frac{1-m}{1 -e^{-(1-m)t}}\n\t\t\\Big(C(1-\\kappa_2)^{-d-\\alpha}+1\\Big) \\frac1{\\delta |\\tilde{y}|^{\\alpha}} \\int_0^\\infty u^2 \\: \\mu_t({\\rm d} u) \\\\\n\t\t&\\qquad\\qquad\\leq \\frac{2 \\big(C(1-\\kappa_2)^{-d-\\alpha}+1\\big)}{\\delta(1-m)|\\tilde{y}|^{\\alpha}}.\n\t\\end{align*}\n\tNow we use either $(1-m)\\geq (1-\\kappa_1^\\alpha)$ or $(1-m)|\\tilde{y}|^{\\alpha}=|y|$. To conclude the proof, it enough to\n\tnotice that by \\eqref{eq:1-moment} the expression\n\t\\[\n\t\t \\frac{(1-m)t}{1 -e^{-(1-m)t}} e^{-t}\\int_0^1 u \\Phi(t, u) {\\: \\rm d} u\n\t\t = 1 - \\frac{(1-m)t}{1 -e^{-(1-m)t}} e^{-t}\n\t\\]\n\tconverges to the stated limits.\n\\end{proof}\n\n\\begin{corollary}\n\t\\label{cor:ius}\n\tUnder the assumptions of Theorem~\\ref{thm:ius} we have\n\t\\begin{equation}\n\t\t\\label{eq:27a}\n\t\t\\lim_{|y|\\to +\\infty} \\frac{\\rho_{\\mathbf{Y}}(y)}{\\nu(y)}= \\frac1{1-m},\n\t\\end{equation}\n\tand\n\t\\begin{equation}\n\t\t\\label{eq:27b}\n\t\t\\rho_{\\mathbf{Y}} \\approx 1 \\land \\nu, \\qquad\\text{on}\\quad \\RR^d.\n\t\\end{equation}\n\tFurthermore, for all $\\delta,r>0$\n\t\\begin{align}\n\t\t\\label{approx:p_rho}\n\t\tp \\approx \\rho_{\\mathbf{Y}}, \\qquad\\text{on}\\quad [\\delta,\\infty)\\times B_r \\times \\RR^d.\n\t\\end{align}\n\\end{corollary}\n\\begin{proof}\n\tThe equality \\eqref{eq:27a} follows from \\eqref{eq:ius-1} by passing with $t$ to infinity and using\n\t\\eqref{representationrho}. The integral representation \\eqref{representationrho} implies that $\\rho_{\\mathbf{Y}}$\n\tis bounded from above by $\\rho_{\\mathbf{Y}}(0)$. Moreover, it is bounded from below by a positive constant, on every\n\tcompact subset of $\\RR^d$. From \\eqref{eq:27a} we easily deduce that $\\rho_{\\mathbf{Y}}(y) \\approx 1 \\land \\nu(y)$ on\n\t$\\RR^d$. Since $\\inf_{y\\in B_{2r}} \\rho_{\\mathbf{Y}}(y) > 0$, it follows from \\eqref{eq:lim_p_t_infty-unif} that there\n\tis $T > 0$ such that $p \\approx \\rho_{\\mathbf{Y}}$ on $[T,\\infty)\\times B_r \\times B_{2r}$\n\tNow, by positivity and continuity, see Lemma~\\ref{lem:p_reg}, $p \\approx \\rho_{\\mathbf{Y}}$ on\n\t$[\\delta,T]\\times B_r \\times B_{2r}$. It remains to investigate\n\t$(t, x, y) \\in [\\delta,\\infty)\\times B_r \\times (\\RR^d \\setminus B_{2r})$. Using the representation \\eqref{eq:rep-p-x}\n\tand \\eqref{approx:is}, there is $C > 0$ such that\n\t\\begin{equation}\n\t\t\\label{eq:86}\n\t\tC^{-1} p(t; 0, y) \\leq p(t;x,y) \\leq C p(t;0,y)\n\t\\end{equation}\n\tfor all $(t, x, y) \\in (0,\\infty)\\times B_r \\times (\\RR^d \\setminus B_{2r})$.\n\tNow, by \\eqref{eq:rep-p-0.1}, \\eqref{ineq:2} and \\eqref{eq:1-moment}, for all $t>0$, $y\\in\\RR^d$,\n\t\\[\n\t\t\\frac{p(t;0,y)}{\\nu(y)} \\leq C \\int_0^\\infty u \\: \\mu_t({\\rm d} u) \\leq \\frac{C}{1-m}\\,.\n\t\\]\n\tUsing again \\eqref{eq:1-moment} and \\eqref{ineq:2-moment}, we obtain that for all $t>0$,\n\t\\begin{align}\n\t\t\\label{ineq:cut-M}\n\t\t\\int_{[M,\\infty)} u \\: \\mu_t({\\rm d} u)\n\t\t\\leq \\frac12 \\int_0^\\infty u \\: \\mu_t({\\rm d} u)\n\t\\end{align}\n\twhere $M= 4/(1-m^2)$. Thus, by \\eqref{approx:is} we have $p_0(s;0,w) \\geq C^{-1} s \\nu(w)$ provided that\n\t$|w| \\geq r (s/M)^{1/\\alpha}$. Hence, by \\eqref{eq:rep-p-0.1}, for all $t \\geq \\delta$, $|y|\\geq 2r$,\n\t\\begin{align*}\n\t\t\\frac{p(t;0,y)}{\\nu(y)} \\geq C^{-1} \\int_{(0,M]} u \\: \\mu_t({\\rm d} u)\n\t\t&\\geq \\frac{C^{-1}}2 \\int_0^\\infty u \\: \\mu_t({\\rm d} u) \\\\\n\t\t&\\geq \\frac{C^{-1}}2 \\frac{1-e^{-(1-m)\\delta}}{1-m},\n\t\\end{align*}\n\twhich together with \\eqref{eq:27b} and \\eqref{eq:86} completes the proof.\n\\end{proof}\n\n\\subsection{$\\alpha$-stable subordinator with $\\alpha\\in (0,1)$ and resetting}\n\\label{sec:sub}\nLet $\\nu$ be the density of the L{\\'e}vy measure of an $\\alpha$-stable subordinator, $\\alpha \\in (0, 1)$. Namely,\n\\[\n\t\\nu(s)=\\frac{\\alpha}{\\Gamma\\left(1-\\alpha\\right)} \\frac{\\ind{(0,\\infty)}(s)}{s^{1+\\alpha}},\n\\]\nsee Example~\\ref{ex:names:1}. A similar result to Theorem \\ref{thm:ius} holds in the present setting as well.\n\\begin{theorem}\n\t\\label{thm:s-s}\n\tSuppose that $\\mathbf{Y}$ is an $\\alpha$-stable subordinator with $\\alpha \\in (0, 1)$. Assume that $\\mathbf{X}^{(c)}$\n\tis obtained from $\\mathbf{Y}$ by partial resetting with factor $c\\in(0,1)$. Then for each $\\kappa_1, \\kappa_2 \\in (0, 1)$,\n\tthe transition density $p^{(c)}$ of $\\mathbf{X}^{(c)}$ satisfies\n\t\\begin{align}\n\t\t\\label{eq:s-s-1}\n\t\t\\lim_{\\atop{y \\to +\\infty}{t \\to +\\infty}} \\, \\sup_{\\substack{ c \\in(0,\\kappa_1] \\\\ |x| \\leq \\kappa_2 y}}\n\t\t\\left| (1-m)\\frac{p^{(c)}(t; x, y)}{\\nu(y)}-1 \\right|= 0\\,.\n\t\\end{align}\n\tFurthermore, for all $K>0$,\n\t\\[\n\t\t\\lim_{\\substack{ y \\to +\\infty \\\\ t \\to+\\infty}}\\,\n\t\t\\sup_{\\substack{c \\in (0,1)\\\\ |x|\\leq K y}}\n\t\t\\left| \\frac{(1-m)t}{1 -e^{-(1-m)t}}\\frac{p^{(c)}(t; x, t^{1/\\alpha} y)}{t\\nu(t^{1/\\alpha}y)} - 1 \\right|\n\t\t= 0,\n\t\\]\n\tand\n\t\\[\n\t\t\\lim_{\\substack{y \\to +\\infty \\\\ t \\to+\\infty} }\n\t\t\\sup_{\\substack{c \\in (0,1)\\\\ |x|\\leq K }}\n\t\t\\left| \\frac{(1-m)t}{1 -e^{-(1-m)t}}\\frac{p^{(c)}(t; x, (1-m)^{-1/\\alpha}y)}{t\\nu((1-m)^{-1/\\alpha}y)} - 1 \\right|\n\t\t= 0.\n\t\\]\t\n\\end{theorem}\n\\begin{proof}\n\tFirst, let us recall that\n\t\\begin{equation}\n\t\t\\label{eq:34}\n\t\t\\lim_{s w^{-\\alpha} \\to 0^+} \\frac{p_0(s;0,w)}{s\\nu(w)}=1,\n\t\\end{equation}\n\tsee e.g. \\cite[Theorem 37.1]{MR0344810} or \\cite[Theorem 2.5.6]{MR854867}. In particular, \\eqref{eq:34} describes the\n\tasymptotic behavior of $p_0(1/\\tilde{c}; 0, y)$ with $\\tilde{c}=\\cos(\\pi\\alpha/2)$, cf. \\cite[Example 24.12]{MR1739520}.\n\tBy the formula \\cite[(9)]{MR2013738} we also have\n\t\\begin{align}\n\t\t\\label{ineq:s-s-2}\n\t\tp_0(s;0,w) \\leq C s \\nu(w).\n\t\\end{align}\n\tNow the arguments follows by the same line of reasoning as the proof of Theorem~\\ref{thm:ius}.\n\\end{proof}\n\n\\begin{corollary}\n\t\\label{cor:s-s}\n\tUnder the assumptions of Theorem~\\ref{thm:s-s} we have\n\t\\begin{equation}\n\t\t\\label{eq:35a}\n\t\t\\lim_{y\\to +\\infty}\n\t\t\\frac{\\rho_{\\mathbf{Y}}(y)}{\\nu(y)}\n\t\t=\n\t\t\\frac1{1-m}.\n\t\\end{equation}\n\tMoreover, for each $\\delta > 0$,\n\t\\begin{equation}\n\t\t\\label{eq:35b}\n\t\t\\rho_{\\mathbf{Y}} \\approx \\nu \\qquad\\text{on } [\\delta, \\infty).\n\t\\end{equation}\n\tFurthermore, for each $\\delta, r>0$,\n\t\\begin{align}\n\t\t\\label{approx:p_rho-2}\n\t\tp \\approx \\rho_{\\mathbf{Y}} \\qquad\\text{on } [\\delta,\\infty)\\times (-r,r)\\times [\\delta,\\infty).\n\t\\end{align}\n\\end{corollary}\n\\begin{proof}\n\tThe equality \\eqref{eq:35a} follows from \\eqref{eq:s-s-1} by passing with $t$ to infinity and using\n\t\\eqref{representationrho}. Since both $\\rho_{\\mathbf{Y}}$ and $\\nu$ are continuous and positive on $(0,\\infty)$, by\n\t\\eqref{eq:35a} we get $\\rho_{\\mathbf{Y}} \\approx \\nu$ for $y \\in [\\delta, \\infty)$. Because\n\t$\\inf_{y\\in (\\delta,2r)} \\rho_{\\mathbf{Y}}(y)>0$, by \\eqref{eq:lim_p_t_infty-unif} there is $T > 0$ such that\n\t$p \\approx \\rho_{\\mathbf{Y}}$ on $[T,\\infty)\\times (-r,r) \\times [\\delta, 2r)$. We notice that $p(t;x,y)>0$ for all\n\t$t>0$, $x\\in\\RR$ and $y>0$, see \\eqref{eq:rep-p-x}. Since $p$ is continuous, see Lemma \\ref{lem:p_reg}, the comparability\n\t$p \\approx \\rho_{\\mathbf{Y}}$ holds on $[\\delta,T] \\times (-r,r) \\times [\\delta,2r)$. Therefore, it remains to consider\n\t$(t, x, y) \\in [\\delta,\\infty)\\times (-r,r)\\times [2r,\\infty)$. Let us observe that for $|x|\\leq r$, and $y \\geq 2r$,\n\t\\[\n\t\t\\nu(y-c^jx)\\leq 2^{1+\\alpha}\\nu(y).\n\t\\]\n\tHence, using the representation \\eqref{eq:rep-p-x} and \\eqref{ineq:s-s-2}, for all\n\t$(t, x, y) \\in [\\delta,\\infty)\\times (-r,r)\\times [2r,\\infty)$, we get\n\t\\begin{align*}\n\t\tp(t;x,y)\n\t\t&\\leq\n\t\t2^{1+\\alpha} \\nu(y)\\, C \\int_0^\\infty u \\: \\mu_t({\\rm d} u) \\\\\n\t\t&\\leq 2^{1+\\alpha}\\frac{C}{1-m} \\nu(y)\n\t\\end{align*}\n\twhere the last estimate is a consequence of \\eqref{eq:1-moment}. In view of the formula \\cite[(10)]{MR2013738}, for\n\t$w \\geq (s/M)^{1/\\alpha}$ with $M=4/(1-m^2)$, and $s > 0$,\n\t\\[\n\t\tp_0(s;0,w) \\geq C^{-1} s \\nu(w).\n\t\\]\n\tMoreover, for $u \\in (0, 1)$,\n\t\\[\n\t\tu -c^jx \\geq r (tu/M)^{1/\\alpha},\n\t\\]\n\tthus, using \\eqref{eq:rep-p-x} together with \\eqref{ineq:cut-M} and \\eqref{eq:1-moment}, we get\n\t\\begin{align*}\n\t\tp(t;x,y)\n\t\t&\\geq\n\t\te^{-t} p_0(t;0,y-x) \\ind{(0,M]}(t)+e^{-t}\\sum_{j=1}^\\infty t^j\n\t\t\\int_0^1 p_0(tu;0,y-c^jx) \\ind{(0,M]}(tu) P_j(u) {\\: \\rm d} u \\\\\n\t\t&\\geq 2^{-1-\\alpha} \\nu(y)\\, C^{-1} \\int_{(0,M]} u \\: \\mu_t({\\rm d} u) \\\\\n\t\t&\\geq 2^{-1-\\alpha} \\nu(y)\\,\\frac{C^{-1}}2 \\int_0^\\infty u \\: \\mu_t({\\rm d} u)\\\\\n\t\t&\\geq 2^{-1-\\alpha} \\frac{C^{-1}}2 \\frac{1-e^{-(1-m)\\delta}}{1-m} \\nu(y),\n\t\\end{align*}\n\twhich together with \\eqref{eq:35b} completes the proof.\n\\end{proof}\n\n\\subsection{$d$-Cylindrical $\\alpha$-stable process with $\\alpha\\in(0,2)$ and resetting}\n\\label{sec:cyl}\nIn this section we study $d$-dimensional cylindrical $\\alpha$-stable process, $\\alpha \\in (0, 2)$, namely a L\\'evy\nprocess with the density\n\\[\n\tp_0(t; x, y) = \\prod_{j=1}^d q(t; x_j, y_j)\n\\]\nwhere $q$ denotes the density of the symmetric one-dimensional $\\alpha$-stable process. Let $\\nu$ be\nthe L{\\'e}vy measure of the one-dimensional process, that is\n\\[\n\t\\nu(r)=\\frac{2^{\\alpha}\\Gamma((1+\\alpha)/2)}{\\pi^{1/2}|\\Gamma(-\\alpha/2)|}\n\t|r|^{-1-\\alpha}, \\qquad r \\in \\RR.\n\\]\nThe function $\\nu$ should not be confused with the L{\\'e}vy measure of the $d$-cylindrical $\\alpha$-stable process,\nwhich is singular, see Example~\\ref{ex:names:3} for details.\n\nGiven $\\theta\\in\\RR^d$ let us define\n\\[\n\tS_1=\\{i \\in \\{1, \\ldots, d\\} : \\theta_i\\neq 0\\}\\,,\\qquad\n\td_1=\\#S_1\\,,\\qquad\n\tS_0=\\{1, \\ldots, d\\} \\setminus S_1.\n\\]\nRecall that $p(t;x,y)$ depends on $c$ (or equivalently on $m$), see \\eqref{def:m}.\n\n\\begin{theorem}\n\t\\label{thm:cylindrical}\n\tSuppose that $\\mathbf{Y}$ is a cylindrical $\\alpha$-stable process in $\\RR^d$, $\\alpha \\in (0, 2)$, with a transition\n\tdensity $p_0$. Assume that $\\mathbf{X}$ is obtained from $\\mathbf{Y}$ by partial resetting with factor $c\\in(0,1)$.\n\tThen for each $\\theta\\in \\RR^d$, $\\kappa\\in (0,1)$, $K\\geq 0$ and $\\delta>0$, the transition density of $\\mathbf{X}$\n\tsatisfies\n\t\\begin{align}\n\t\t\\label{eq:c-1}\n\t\t\\lim_{\\substack{ r \\to +\\infty \\\\ t \\to+\\infty}}\n\t\t\\sup_{y \\in \\mathscr{Y}} \\sup_{x\\in \\mathscr{X}(r,t)}\n\t\t\\bigg| \\frac{p(t;x,r\\theta+y)}{\\prod_{i\\in S_1}\\nu(r\\theta_i)}- L \\bigg|=0\n\t\\end{align}\n\twhere\n\t\\begin{align*}\n\t\t\\mathscr{X}(r,t)\n\t\t&=\n\t\t\\left\\{x\\in\\RR^d :\n\t\t\\begin{aligned}\n\t\t\t\\abs{x_i} &\\leq \\kappa \\norm{r \\theta_i}, \\quad \\text{for all}\\quad i \\in S_1, \\\\\n\t\t\t\\log_{1/c} \\abs{x_i} &\\leq \\frac{t}{1+\\delta}, \\quad \\text{for all}\\quad i \\in S_0,\n\t\t\\end{aligned}\n\t\t\\right\\} \\\\\n\t\t\\mathscr{Y}&=\\Big\\{y\\in\\RR^d : |y_i|\\leq K \\quad \\text{for all}\\quad i \\in S_1\\Big\\},\n\t\\end{align*}\n\tand\n\t\\[\n\t\tL= \\int_0^{\\infty} u^{d_1} \\prod_{i\\in S_0} q(u;0,y_i) \\: \\mu({\\rm d}u)\\,.\n\t\\]\n\\end{theorem}\n\\begin{proof}\n\tLet $d_0=\\#S_0$. For $s > 0$ and $w \\in \\RR^d$,\n\t\\begin{align*}\n\t\tp_0(s;0,w)=\n\t\t\\bigg\\{ \\prod_{i\\in S_1} q(s;0,w_i) \\bigg\\}\n\t\t\\bigg\\{\\prod_{i\\in S_0} q(s;0,w_i)\\bigg\\}.\n\t\\end{align*}\n\tRecall that for all $s>0$, $v\\in\\RR$, (see \\cite{BlumenthalGetoor})\n\t\\begin{align}\n\t\t\\label{approx:is-c}\n\t\tq(s;0,v) \\approx \\min\\bigg\\{ s^{-d/\\alpha}, \\frac{s}{|v|^{1+\\alpha}}\\bigg\\}.\n\t\\end{align}\n\tIn particular,\n\t\\begin{align}\n\t\t\\label{ineq:2-c}\n\t\tq(s;0,v) & \\leq C \\min\\big\\{ s^{-1/\\alpha}, s \\nu(v)\\big\\}.\n\t\\end{align}\n\tUsing \\eqref{eq:rep-p-x} we write\n\t\\begin{align*}\n\t\tI_1 &= \\frac{p(t; x, r\\theta + y)}{\\prod_{i\\in S_1}\\nu(r\\theta_i)} \\\\\n\t\t&=\n\t\te^{-t} t^{d_1} \\frac{p_0(t;0,r\\theta+y-x)}{\\prod_{i\\in S_1}t \\nu(r\\theta_i)}\n\t\t+ e^{-t}\n\t\t\\sum_{j = 1}^\\infty t^j \\int_0^1 \\frac{p_0(t u; 0,r\\theta+ y-c^j x)}{\\prod_{i\\in S_1} (tu)\n\t\t\\nu(r\\theta_i)}(tu)^{d_1} P_j(u) {\\: \\rm d} u.\n\t\\end{align*}\n\tFirst, we show that\n\t\\[\n\t\t\\lim_{\\substack{ r \\to +\\infty \\\\ t \\to+\\infty}}\n\t\t\\sup_{y \\in \\mathscr{Y}} \\sup_{x\\in \\mathscr{X}(r,t)} \\abs{I_1-I_2}=0\n\t\\]\n\twhere\n\t\\[\n\t\tI_2=e^{-t}f(t,x)+e^{-t}\\sum_{j=1}^\\infty t^j \\int_0^1 f(tu,c^jx) P_j(u) \\: {\\rm d}u,\n\t\\]\n\tand\n\t\\[\n\t\tf(u,x)= u^{d_1} \\prod_{i\\in S_0} q(u;0,y_i-x_i).\n\t\\]\n\tLet\n\t\\[\n\t\tr > 2 \\frac{K}{1-\\kappa}\\min\\big\\{\\abs{\\theta_i}^{-1} : i \\in S_1\\big\\}.\n\t\\]\n\tSince for $i \\in S_1$,\n\t\\begin{equation}\n\t\t\\label{eq:71}\n\t\t|x_i| \\leq \\kappa r|\\theta_i|, \\qquad\\text{ and }\\qquad r |\\theta_i| \\geq |y_i| \\frac{2}{1-\\kappa},\n\t\\end{equation}\n\tfor each $j \\in \\NN_0$, we have\n\t\\begin{align}\n\t\t\\label{ineq:1-c}\n\t\t\\frac{\\nu(r\\theta_i+y_i-c^jx_i)}{\\nu(r\\theta_i)}\n\t\t\\leq \\left(\\frac{1-\\kappa}{2}\\right)^{-1-\\alpha}.\n\t\\end{align}\n\tNow using \\eqref{ineq:2-c} and \\eqref{ineq:1-c} we obtain\n\t\\[\n\t\te^{-t} t^{d_1} \\frac{p_0(t;0,r\\theta+y-x)}{\\prod_{i\\in S_1} t \\nu(r\\theta_i)}\n\t\t\\leq\n\t\t\\tilde{C}^{d_1} C^{d_0} e^{-t} t^{d_1-d_0/\\alpha},\n\t\\]\n\tand\n\t\\[\n\t\te^{-t}f(t,x) \\leq C^{d_0} e^{-t} t^{d_1-d_0/\\alpha}\n\t\\]\n\twhere $\\tilde{C} = 2^{1+\\alpha} C (1-\\kappa)^{-1-\\alpha}$. Hence, the first terms in $I_1$ as well as in $I_2$ are\n\tnegligible. Furthermore, by \\eqref{ineq:2-c} and \\eqref{ineq:1-c} for $j\\in\\NN$ we have\n\t\\begin{align}\n \t\te^{-t} t^j\n\t\t\\int_0^1\n\t\t\\bigg( \\frac{p_0(tu;0,r\\theta+y-c^j x)}{\\prod_{i\\in S_1} (tu)\\nu(r\\theta_i)}(tu)^{d_1}+ f(tu,c^j x)\\bigg)\n\t\tP_j(u) {\\: \\rm d} u \\nonumber \\\\\n\t\t\\leq\n\t\t\\big( \\tilde{C}^{d_1}+1\\big) C^{d_0} e^{-t} t^j\n\t\t\\int_0^1 (tu)^{d_1-d_0/\\alpha} P_j(u) {\\: \\rm d} u. \\label{ineq:b-2-c}\n\t\\end{align}\n\tLet us recall that (see e.g. \\cite{BlumenthalGetoor})\n\t\\begin{equation}\n\t\t\\label{eq:72}\n\t\t\\lim_{s /|v|^{\\alpha} \\to 0^+} \\frac{q(s; 0, v)}{s \\nu(v)} = 1.\n\t\\end{equation}\n\tFor $j \\in \\NN$, we write\n\t\\begin{equation}\n\t\t\\label{eq:75}\n\t\t\\begin{aligned}\n\t\t\\bigg|\\frac{q(s; 0,r\\theta_i+ y_i-c^j x_i)}{s \\nu(r\\theta_i)} - 1 \\bigg|\n\t\t&\\leq\n\t\t\\bigg|\\frac{q(s; 0, r\\theta_i+ y_i-c^j x_i)}{s \\nu(r\\theta_i+ y_i-c^j x_i)} - 1 \\bigg|\n\t\t\\frac{\\nu(r\\theta_i+ y_i-c^j x_i)}{\\nu(r\\theta_i)}\\\\\n\t\t&\\phantom{\\leq}+\\bigg|\\frac{\\nu(r\\theta_i+ y_i-c^j x_i)}{\\nu(r\\theta_i)} -1\\bigg|.\n\t\t\\end{aligned}\n\t\\end{equation}\n\tBy \\eqref{eq:71}, for $i \\in S_1$ and $j \\in \\NN$,\n\t\\begin{equation}\n\t\t\\label{eq:73}\n\t\t\\frac{s}{|r\\theta_i +y_i-c^jx_i|^{\\alpha}}\n\t\t\\leq\n\t\t2^{\\alpha} (1 - \\kappa)^{-\\alpha}\n\t\t\\frac{s}{\\abs{r \\theta_i}^\\alpha},\n\t\\end{equation}\n\tand\n\t\\begin{equation}\n\t\t\\label{eq:74}\n\t\t\\frac{\\nu(r\\theta_i)}{\\big(1+K/|r\\theta_i|+c^j\\kappa\\big)^{1+\\alpha}}\n\t\t\\leq\n\t\t\\nu(r\\theta_i+ y_i-c^j x_i)\n\t\t\\leq\n\t\t\\frac{\\nu(r\\theta_i)}{\\big(1-K/|r\\theta_i|-c^j\\kappa\\big)^{1+\\alpha}}.\n\t\\end{equation}\n\tLet $\\epsilon>0$. By \\eqref{eq:75}, in view of \\eqref{eq:72}, \\eqref{eq:73} and \\eqref{eq:74}, we deduce that\n\tthere are $\\tilde{\\delta} > 0$ and $n_0 \\in \\NN$, such that if $s r^{-\\alpha} \\leq \\tilde{\\delta}$, then\n\tfor all $j > n_0$, $r \\geq (\\tilde{\\delta})^{-1}$ and $i \\in S_1$,\n\t\\begin{equation}\n\t\t\\label{ineq:3-c}\n\t\t\\bigg|\\frac{q(s; 0, r\\theta_i+ y_{0.i}-c^j xi)}{s \\nu(r\\theta_i)} - 1 \\bigg| \\leq \\epsilon.\n\t\\end{equation}\n\tBoth $\\tilde{\\delta}$ and $n_0$ depend only on $\\alpha$, $\\epsilon$, $\\kappa$, $K$ and $\\theta$. Now, we write\n\t\\begin{equation}\n\t\t\\label{eq:77}\n\t\t\\begin{aligned}\n\t\t&e^{-t} \\sum_{j = 1}^\\infty t^j\n\t\t\\int_0^1 \\frac{p_0(t u; 0,r\\theta+ y-c^j x)}{\\prod_{i\\in S_1} (tu) \\nu(r\\theta_i)}(tu)^{d_1} P_j(u) {\\: \\rm d} u\n\t\t-e^{-t}\\sum_{j=1}^\\infty t^j \\int_0^1 f(tu,c^jx) P_j(u) \\: {\\rm d}u\\\\\n\t\t&\\qquad= e^{-t} \\sum_{j=1}^{n_0} t^j \\int_0^1\n\t\t\\bigg( \\frac{p_0(tu;0,r\\theta+y-c^j x)}{\\prod_{i\\in S_1} (tu)\\nu(r\\theta_i)}(tu)^{d_1} - f(tu,c^j x)\\bigg) P_j(u)\n\t\t{\\: \\rm d} u\\\\\n\t\t&\\qquad\\phantom{=}\n\t\t+e^{-t} \\sum_{j=n_0+1}^\\infty t^j\\int_{\\frac{\\tilde{\\delta} r^\\alpha}{t}\\land 1}^1\n\t\t\\bigg( \\frac{p_0(tu;0,r\\theta+y-c^j x)}{\\prod_{i\\in S_1} (tu)\\nu(r\\theta_i)}(tu)^{d_1} - f(tu,c^j x) \\bigg)\n\t\tP_j(u) {\\: \\rm d} u \\\\\n\t\t&\\qquad\\phantom{=}\n\t\t+ e^{-t} \\sum_{j=n_0+1}^\\infty t^j\n\t\t\\int_0^{\\frac{\\tilde{\\delta} r^\\alpha}{t}\\land 1}\n\t\t\\bigg( \\frac{p_0(tu;0,r\\theta+y-c^j x)}{\\prod_{i\\in S_1} (tu)\\nu(r\\theta_i)}(tu)^{d_1} - f(tu,c^j x)\\bigg)\n\t\tP_j(u) {\\: \\rm d} u.\n\t\t\\end{aligned}\n\t\\end{equation}\n\tBy \\eqref{ineq:b-2-c}, the first term on the right-hand side of \\eqref{eq:77} is small, see also Theorem\n\t\\ref{thm:all-moments}. The absolute value of the third term can be bounded by\n\t\\begin{align*}\n\t\te^{-t}\n\t\t\\sum_{j = n_0+1}^\\infty t^j\n\t\t\\int_0^{\\frac{\\tilde{\\delta} r^\\alpha}{t} \\land 1}\n\t\t\\bigg| \\prod_{i\\in S_1} \\frac{q(tu;0,r\\theta_i+y_i-c^jx_i)}{tu \\nu(r\\theta_i)} - 1\\bigg|\n\t\t\\bigg( \\prod_{i\\in S_0} q(tu;0,y_i-c^jx_i)\\bigg) (tu)^{d_1} P_j(u) {\\: \\rm d} u.\n\t\\end{align*}\n\tIn view of \\eqref{ineq:2-c}, \\eqref{ineq:1-c} and \\eqref{ineq:3-c} the last displayed formula is bounded by\n\t\\begin{align*}\n\t\t\\epsilon e^{-t}\n\t\t\\sum_{j = 1}^\\infty t^j \\int_0^1 \\big(\\tilde{C}\\vee 1 \\big)^{d_1-1} C^{d_0} (tu)^{d_1-d_0/\\alpha} P_j(u) {\\: \\rm d} u\n\t\t\\leq\n\t\t\\epsilon \\big( \\tilde{C}\\vee 1 \\big)^{d_1-1} C^{d_0} \\sup_{t\\geq 1} \\int_0^\\infty u^{d_1-d_0/\\alpha} \\mu_t({\\rm d}u)\n\t\\end{align*}\n\twhich is small by \\eqref{ineq:sup-finite}. Next, using \\eqref{ineq:2-c} and \\eqref{ineq:1-c}, the absolute value of the\n\tsecond term in \\eqref{eq:77} is controlled as follows\n\t\\begin{align*}\n\t\t&e^{-t}\n\t\t\\sum_{j=1}^\\infty\n\t\tt^j\\int_{\\frac{\\tilde{\\delta} r^\\alpha}{t}\\land 1}^1\n\t\t\\bigg( \\frac{p_0(tu;0,r\\theta+y-c^j x)}{\\prod_{i\\in S_1} (tu)\\nu(r\\theta_i)}(tu)^{d_1} + f(tu,c^j x)\\bigg)\n\t\tP_j(u) {\\: \\rm d} u\\\\\n\t\t&\\qquad\n\t\t\\leq\n\t\t\\big(\\tilde{C}^{d_1}+1\\big) C^{d_0} \\int_{\\tilde{\\delta} r^\\alpha \\land t}^t\n\t\tu^{d_1-d_0/\\alpha} \\mu_t({\\rm d}u)\\\\\n\t\t&\\qquad\n\t\t\\leq \\big( \\tilde{C}^{d_1}+1\\big) C^{d_0}\n\t\t\\frac1{\\tilde{\\delta} r^\\alpha}\n\t\t\\sup_{t\\geq 1} \\int_0^\\infty u^{d_1-d_0/\\alpha+1} \\mu_t({\\rm d}u).\n\t\\end{align*}\n\tConsequently, $|I_1-I_2|$ converges to zero as desired.\n\t\n\tTherefore, it suffices to show that\n\t\\[\n\t\t\\lim_{\\substack{r \\to +\\infty \\\\ t \\to +\\infty}} \\sup_{y \\in \\mathscr{Y}} \\sup_{x \\in \\mathscr{X}(r, t)}\n\t\t\\big|I_2 - L \\big| = 0.\n\t\\]\n\tWithout loss of generality we assume that $S_0=\\{1,\\ldots, d_0\\}$. Since $\\prod_{i\\in S_0} q(s;0,w_i)$ may be regarded as\n\ta transition density of a strictly $\\alpha$-stable process in $\\RR^{d_0}$, it follows from Lemma~\\ref{lem:A3} that\n\tthe family\n\t\\[\n\t\t\\mathcal{F}=\\Big\\{ f : \\RR_+ \\times \\RR^d \\rightarrow \\RR :\n\t\tf(u, x) = u^{d_1} \\prod_{i\\in S_0} q(u;0,y_i-x_i), \\, y\\in \\RR^{d_0} \\Big\\}\n\t\\]\n\tsatisfies \\eqref{eq:78}. Notice that for each $f \\in \\mathcal{F}$ we have $\\int_0^\\infty f(u,0)\\: \\mu({\\rm d} u)=L$.\n\tApplying now Lemma \\ref{lem:unif_conv-2} with $\\delta$ replaced by $\\delta/2$, we get\n\t\\[\n\t\t\\lim_{t\\to +\\infty} \\sup_{f\\in \\mathcal{F}} \\sup_{x \\in \\mathscr{X}_{\\delta/2}(t)} |I_2-L|=0.\n\t\\]\n\tIn particular, $I_2 \\rightarrow L$ uniformly with respect to $y \\in \\RR^{d_0}$. Furthermore, since\n\t$x \\in \\mathscr{X} (r,t)$ implies that $x \\in \\mathscr{X}_{\\delta/2}(t)$ provided that\n\t\\[\n\t\tt \\geq (1+\\delta/2)(1+\\delta^{-1}) \\log_{1/c} d_0,\n\t\\]\n\tthe theorem follows.\n\\end{proof}\n\nIn addition to studying the asymptotic behavior of $p$, we also show its upper and lower estimates.\n\\begin{proposition}\n\t\\label{prop:cylindrical}\n\tUnder the assumptions of Theorem~\\ref{thm:cylindrical}, for all $\\theta\\in\\RR^d$ and $K\\geq 0$,\n\t\\begin{equation}\n\t\t\\label{eq:79}\n\t\t\\lim_{r\\to\\infty} \\sup_{y\\in\\mathscr{Y}}\n\t\t\\bigg| \\frac{\\rho_{\\mathbf{Y}}(r\\theta+y)}{\\prod_{i\\in S_1}\\nu(r\\theta_i)} - L\\bigg|=0 \\,.\n\t\\end{equation}\n\tFurthermore, for all $\\delta, r>0$,\n\t\\begin{equation}\n\t\t\\label{eq:80}\n\t\tp \\approx \\rho_{\\mathbf{Y}} \\qquad\\text{on } [\\delta,\\infty) \\times B_r \\times \\RR^d,\n\t\\end{equation}\n\tand\n\t\\[\n\t\t\\rho_{\\mathbf{Y}}(y) \\approx \\prod_{i=1}^d 1\\land \\nu(y_i) \\qquad\\text{for } y \\in \\RR^d.\n\t\\]\n\\end{proposition}\n\\begin{proof}\n\tTo get \\eqref{eq:79}, it is enough to use \\eqref{representationrho} and take $t$ tending to infinity in \\eqref{eq:c-1}.\n\tNow, to prove \\eqref{eq:80}, it is enough to show that\n\t\\[\n\t\t\\frac{p(t;x,y)}{\\prod_{i=1}^d 1\\land \\nu(y_i)} \\approx 1,\n\t\t\\qquad\\text{on } [\\delta,\\infty)\\times B_r \\times \\RR^d.\n\t\\]\n\tWithout loss of generality we assume that $\\delta < 1$. Given $y\\in\\RR^d$ and $r > 0$ we set\n\t\\begin{align*}\n\t\tS_1(y, r)&=\\{i \\in \\{1, \\ldots, d\\} : |y_i|> 2r\\}, &\\qquad\n\t\td_1(y, r)&=\\#S_1(y, r), \\\\\n\t\tS_0(y, r)&=\\{i \\in \\{1, \\ldots, d\\} : |y_i|\\leq 2r\\}, &\\qquad\n\t\td_0(y, r)&=\\#S_0(y, r).\n\t\\end{align*}\n\tThe reasoning is analogous to the proof of Theorem \\ref{thm:cylindrical}. For $s > 0$ and $y, w \\in \\RR^d$, we write\n\t\\[\n\t\tp_0(s;0,y-w)= \\bigg\\{ \\prod_{i\\in S_1(y, r)} q(s;0,y_i-w_i) \\bigg\\}\n\t\t\\bigg\\{\\prod_{i\\in S_0(y, r)} q(s;0,y_i-w_i)\\bigg\\}.\n\t\\]\n\tLet $t \\geq \\delta$ and $r \\geq \\norm{x}$. For $y \\in \\RR^d$, by \\eqref{ineq:2-c}, we have\n\t\\begin{align*}\n\t\t\\frac{q(s;0,y_i-c^jx_i)}{s \\nu(y_i)} &\\leq 2^{1+\\alpha} C,\n\t\t\\qquad\\text{ for all } i \\in S_1(y, r),\n\t\t\\intertext{and}\n\t\tq(s;0,y_i-c^jx_i) &\\leq C s^{-1/\\alpha}, \\qquad\\text{ for all } i \\in S_0(y, r),\n\t\\end{align*}\n\thence by \\eqref{eq:rep-p-x}\n\t\\begin{align*}\n\t\tp(t; x,y)\n\t\t&\\leq\n\t\t2^{d_1(y, r) (1+\\alpha)} C^d\n\t\t\\bigg(\\prod_{i\\in S_1(y, r)} \\nu(y_i)\\bigg) \\int_0^\\infty u^{d_1(y,r)-d_0(y,r)/\\alpha} \\: \\mu_t({\\rm d} u).\n\t\\end{align*}\n\tSince the latter integral is bounded by the sum of moments of $\\mu_t$ of orders $-d/\\alpha$ and $d$, using\n\t\\eqref{ineq:sup-finite} we obtain the desired upper bound.\n\n\tNext, \\eqref{approx:is-c} implies that\n\t\\begin{align*}\n\t\tq(s;,0,v) &\\geq C^{-1} s \\nu(v), \\qquad\\text{if } |v|\\geq r s^{1/\\alpha},\n\t\t\\intertext{and}\n\t\tq(s;,0,v)&\\geq C^{-1}, \\qquad \\text{if } \\norm{v} \\leq 3r, s \\in [\\delta/2,1].\n\t\\end{align*}\n\tTherefore, for $s \\in [\\delta/2,1]$,\n\t\\begin{align*}\n\t\t\\frac{q(s;0,y_i-c^jx_i)}{s\\nu(y_i)} &\\geq 2^{-1-\\alpha}C^{-1},\n\t\t\\qquad\\text{for all } i \\in S_1(y, r),\n\t\t\\intertext{and}\n\t\tq(s;0,y_i-c^jx_i) &\\geq C^{-1}, \\qquad\\text{for all } i \\in S_0(y, r).\n\t\\end{align*}\n\tTherefore, by \\eqref{eq:rep-p-x} we get\n\t\\begin{align*}\n\t\tp(t;x,y)\n\t\t&\\geq\n\t\te^{-t} p_0(t;0,y-x) \\ind{[\\delta/2,1]}(t)+\n\t\te^{-t}\\sum_{j=1}^\\infty t^j \\int_0^1 p_0(tu;0,y-c^jx)\n\t\t\\ind{[\\delta/2,1]}(tu) P_j(u) {\\: \\rm d} u\\\\\n\t\t&\\geq 2^{-d_1(1+\\alpha)} C^{-d} \\bigg(\\prod_{i\\in S_1(y, r) } \\nu(y_i)\\bigg)\n\t\t\\int_{\\delta/2}^1 u^{d_1(y, r)} \\mu_t({\\rm d}u)\\\\\n\t\t&\\geq 2^{-d_1(y, r) (1+\\alpha)} C^{-d} (\\delta/2)^d \\bigg(\\prod_{i\\in S_1(y, r)}\n\t\t\\nu(y_i)\\bigg)\n\t\t\\inf_{t \\in [t_0,\\infty)} \\mu_t\\big([\\delta/2,1]\\big).\n\t\\end{align*}\n\tTo conclude the proof, we invoke Lemma~\\ref{lem:inf_mu_t}.\n\\end{proof}\n\n\\subsection{Brownian motion with resetting}\nIn this section we consider the case where $\\mathbf{Y}$ is a Brownian motion with a transition density\n\\begin{align}\n\t\\label{p0gauss}\n\tp_0(t;x,y)=(4\\pi t)^{-\\frac{d}{2}} e^{-\\frac{|y-x|^2}{4t}}.\n\\end{align}\nWe are going to study the asymptotic behavior of $p(t; 0, y)$ when $t$ tends to infinity while $(t, y)$ stays in\ncertain space-time regions. By Corollary \\ref{cor:rep-2} and \\eqref{p0gauss}, we have\n\\begin{align}\n\t\\label{eq:p_gauss}\n\tp(t; 0, y) = e^{-t} (4\\pi t)^{-\\frac{d}{2}} e^{-\\frac{|y|^2}{4t}}\n\t+\n\tt e^{-t}(4\\pi t)^{-\\frac{d}{2}} \\int_0^1 u^{-\\frac{d}{2}} e^{-\\frac{|y|^2}{4 tu}+\\log \\frac{\\Phi(t, u)}{t}} {\\: \\rm d} u.\n\\end{align}\nLet us define\n\\[\n\t\\phi(t) = \\sum_{j = 0}^\\infty \\frac{1}{j!} \\frac{1}{(m; m)_{j+1}} t^j, \\qquad t \\geq 0.\n\\]\nIn view of \\eqref{def:Phi} and Proposition \\ref{prop:2}\n\\begin{equation}\n\t\\label{eq:29}\n\t\\Phi(t, u) \\leq t \\phi(t(1-u)), \\qquad \\text{for all } t >0,\\, u \\in [0, 1],\n\\end{equation}\nand\n\\begin{equation}\n\t\\label{eq:30}\n\t\\Phi(t, u) = t \\phi(t(1-u)), \\qquad \\text{for all } t > 0,\\, u \\in [m, 1].\n\\end{equation}\nWe first show a few key properties of the function $\\phi(t)$. Note that for $k\\in\\NN_0$,\n\\begin{equation}\n\t\\label{eq:31}\n\t\\phi^{(k)}(t) = \\sum_{j = 0}^\\infty \\frac{t^j}{j!} \\frac{1}{(m; m)_{j+k+1}}, \\qquad t \\geq 0.\n\\end{equation}\n\n\\begin{proposition}\n\t\\label{prop:7}\n\tFor each $k \\in \\NN_0$, and $t > 0$,\n\t\\begin{equation}\n\t\t\\label{eq:10}\n\t\t\\frac{e^t}{(m; m)_{k+1}} \\leq \\phi^{(k)}(t) \\leq \\frac{e^t}{(m; m)_\\infty},\n\t\\end{equation}\n\tand\n\t\\begin{equation}\n\t\t\\label{eq:11}\n\t\t\\phi(t) \\leq \\phi^{(k)}(t) \\leq \\frac{1}{(m; m)_k} \\phi(t).\n\t\\end{equation}\n\\end{proposition}\n\\begin{proof}\n\tSince\n\t\\[\n\t\t\\frac{1}{(m; m)_{k+1}} \\leq \\frac{1}{(m; m)_{j+k+1}} \\leq \\frac{1}{(m; m)_{\\infty}},\n\t\\]\n\twe easily get \\eqref{eq:10}. Moreover,\n\t\\[\n\t\t\\frac{1}{(m; m)_{j+1}}\n\t\t\\leq\n\t\t\\frac{1}{(m; m)_{j+k+1}}\n\t\t=\n\t\t\\frac{1}{(m; m)_{j+1}} \\prod_{\\ell = 0}^{k-1} \\frac{1}{1-m^{j+2+\\ell}}\n\t\t\\leq\n\t\t\\frac{1}{(m; m)_{j+1}} \\frac{1}{(m; m)_{k}}\n\t\\]\n\twhich leads to \\eqref{eq:11}.\n\\end{proof}\n\n\\begin{proposition}\n\t\\label{prop:8}\n\tFor each $n \\in \\NN$ and $k \\in \\NN$,\n\t\\begin{equation}\n\t\t\\label{lim:phi-1}\n\t\t\\lim_{t \\to \\infty} t^n \\frac{\\phi^{(k+1)}(t) - \\phi^{(k)}(t)}{\\phi^{(k)}(t)} = 0,\n\t\\end{equation}\n\tand\n\t\\begin{align}\n\t\t\\label{lim:phi-2}\n\t\t\\lim_{t \\to \\infty} t^n \\Big(\\frac{\\phi(t)}{e^t} - \\frac1{(m;m)_\\infty}\\Big) = 0.\n\t\\end{align}\n\\end{proposition}\n\\begin{proof}\n\tUsing \\eqref{eq:31}, we get\n\t\\[\n\t\t\\phi^{(k+1)}(t) = \\sum_{j = 0}^\\infty \\frac{t^j}{j!} \\frac{1}{(m; m)_{j+k+1}} \\frac{1}{1-m^{j+k+2}}.\n\t\\]\n\tThus\n\t\\begin{align*}\n\t\t\\phi^{(k+1)}(t)\n\t\t&= \\sum_{j = 0}^\\infty\n\t\t\\frac{t^j}{j!} \\frac{1}{(m; m)_{j+k+1}}\n\t\t\\sum_{\\ell = 0}^\\infty m^{(j+k+2)\\ell} \\\\\n\t\t&=\\sum_{\\ell=0}^\\infty m^{(k+2)\\ell} \\phi^{(k)}\\big(m^\\ell t\\big).\n\t\\end{align*}\n\tBy Proposition \\ref{prop:7}, for $\\ell \\in \\NN$,\n\t\\[\n\t\t\\frac{\\phi^{(k)}(m^\\ell t)}{\\phi^{(k)}(t)} \\leq \\frac{(m; m)_{k+1}}{(m; m)_\\infty} e^{-(1-m^\\ell) t},\n\t\\]\n\thence\n\t\\[\n\t\t\\lim_{t \\to +\\infty} t^n \\frac{\\phi^{(k)}(m^\\ell t)}{\\phi^{(k)}(t)} = 0.\n\t\\]\n\tSince for each $\\ell\\in\\NN$,\n\t\\begin{align*}\n\t\tt^n \\frac{\\phi^{(k)}(m^\\ell t)}{\\phi^{(k)}(t)}\n\t\t&\\leq \\frac{(m; m)_{k+1}}{(m; m)_\\infty} n! \\big(1-m^\\ell\\big)^{-n} \\\\\n\t\t&\\leq \\frac{(m; m)_{k+1}}{(m; m)_\\infty} n! (1-m)^{-n},\n\t\\end{align*}\n\tby the Lebesgue's dominated convergence theorem\n\t\\[\n\t\t\\lim_{t \\to +\\infty} \\sum_{\\ell = 1}^\\infty m^{(k+2)\\ell} t^n \\frac{\\phi^{(k)}(m^\\ell t)}{\\phi^{(k)}(t)} = 0,\n\t\\]\n\twhich completes the proof of \\eqref{lim:phi-1}.\t\n\t\n\tNext, we write\n\t\\[\n\t\t\\frac{1}{(m; m)_\\infty} e^{t} - \\phi(t) = \\sum_{j = 0}^\\infty \\frac{t^j}{j!} \\frac{1}{(m; m)_\\infty}\n\t\t\\Big(1 - \\prod_{k = j+2}^\\infty (1-m^k)\\Big).\n\t\\]\n\tBy the generalized Bernoulli's inequality,\n\t\\[\n\t\t1 - \\prod_{k = j+2}^\\infty (1 - m^k) \\leq \\sum_{k = j+2}^\\infty m^k,\n\t\\]\n\thence\n\t\\begin{align*}\n\t\t\\frac{1}{(m; m)_\\infty} e^{t} - \\phi(t)\n\t\t&\\leq\n\t\t\\sum_{j = 0}^\\infty \\frac{t^j}{j!} \\frac{1}{(m; m)_\\infty} \\sum_{k = j+2}^\\infty m^k \\\\\n\t\t&\\leq\n\t\t\\frac{1}{(m; m)_\\infty} \\frac{m^2}{1-m} e^{m t}.\n\t\\end{align*}\n\tLastly,\n\t\\begin{align*}\n\t\t0 \\leq \\Big(\\frac{1}{(m; m)_\\infty} e^{t} - \\phi(t)\\Big)e^{-t}\n\t\t\\leq \\frac{1}{(m; m)_\\infty} \\frac{m^2}{1-m}\n\t\te^{-(1-m)t}\n\t\\end{align*}\n\twhich completes the proof of \\eqref{lim:phi-2}.\n\\end{proof}\n\n\\begin{lemma}\n\t\\label{lem:1}\n\tFor all $t \\geq 0$,\n\t\\[\n\t\t\\frac{\\phi''(t)}{\\phi(t)} - \\bigg(\\frac{\\phi'(t)}{\\phi(t)} \\bigg)^2 < 0.\n\t\\]\n\\end{lemma}\n\\begin{proof}\n\tUsing \\eqref{eq:31}, for $s,t\\geq 0$, we obtain\n\t\\[\n\t\t\\phi'(s) \\phi(t) - \\phi'(t) \\phi(s)\n\t\t=\n\t\t\\sum_{k = 0}^\\infty \\sum_{\\stackrel{j_1 + j_2 = k}{0 \\leq j_1, j_2}}\n\t\t\\frac{s^{j_1} t^{j_2}}{j_1! j_2!} \\frac{1}{(m; m)_{j_1+1}}\n\t\t\\frac{1}{(m; m)_{j_2+1}} \\Big(\\frac{1}{1-m^{j_1+2}} - \\frac{1}{1-m^{j_2+2}}\\Big).\n\t\\]\n\tObserve that we can assume that $j_1 \\neq j_2$. If $0 \\leq j_1 < j_2$, then\n\t\\[\n\t\t\\frac{1}{1-m^{j_1+2}} - \\frac{1}{1-m^{j_2+2}} = \\frac{m^{j_1+2} - m^{j_2+2}}{(1-m^{j_1+2}) (1-m^{j_2+2})} > 0.\n\t\\]\n\tHence,\n\t\\begin{align*}\n\t\t&\n\t\t\\sum_{\\stackrel{j_1 + j_2 = k}{0 \\leq j_1, j_2}}\n\t\t\\frac{s^{j_1} t^{j_2}}{j_1! j_2!} \\frac{1}{(m; m)_{j_1+1}}\n\t\t\\frac{1}{(m; m)_{j_2+1}} \\Big(\\frac{1}{1-m^{j_1+2}} - \\frac{1}{1-m^{j_2+2}}\\Big) \\\\\n\t\t&\\qquad=\n\t\t\\sum_{\\stackrel{j_1 + j_2 = k}{0 \\leq j_1 < j_2}}\n\t\t\\frac1{j_1! j_2!} \\frac{1}{(m; m)_{j_1+1}} \\frac{1}{(m; m)_{j_2+1}}\n\t\t\\Big(\\frac{1}{1-m^{j_1+2}} - \\frac{1}{1-m^{j_2+2}}\\Big)\n\t\t\\Big(s^{j_1} t^{j_2} - s^{j_2} t^{j_1}\\Big).\n\t\\end{align*}\n\tTherefore, $\\phi''(s) \\phi(t) - \\phi'(t) \\phi'(s)$ equals\n\t\\begin{align*}\n\t\t\\sum_{k = 1}^\\infty\n\t\t\\sum_{\\stackrel{j_1 + j_2 = k}{0 \\leq j_1 < j_2}}\n\t\t\\frac1{j_1! j_2!} \\frac{1}{(m; m)_{j_1+1}} \\frac{1}{(m; m)_{j_2+1}}\n\t\t\\Big(\\frac{1}{1-m^{j_1+2}} - \\frac{1}{1-m^{j_2+2}}\\Big)\n\t\t\\Big(j_1 s^{j_1-1} t^{j_2} - j_2 s^{j_2-1} t^{j_1}\\Big).\n\t\\end{align*}\n\tNow, to conclude the proof it suffices to put $s=t$ and use that $j_1<j_2$.\n\\end{proof}\n\n\\begin{proposition}\n\t\\label{prop:log_phi}\n\tFor each $k, n \\in \\NN$, $k \\geq 2$, there is $C_{n, k} > 0$ such that for all $t \\geq 0$,\n\t\t\\[\n\t\t\tt^n \\Big| \\frac{{\\rm d}^k}{{\\rm d} t^k} \\log \\phi(t) \\Big| \\leq C_{n, k}.\n\t\t\\]\n\\end{proposition}\n\\begin{proof}\n\tLet us recall that for a positive smooth real function $f$, by Fa\\'a di Bruno's formula, there are positive constants\n\t$C_{\\ell}$ such that\n\t\\[\n\t\t\\frac{{\\rm d}^k }{{\\rm d} t^k} \\log f(t) = \\sum_{j = 1}^k j! (-1)^{j+1} \\frac{1}{(f(t))^j}\n\t\t\\sum_{\\ell} C_\\ell \\prod_{i = 1}^j \\frac{{\\rm d}^{\\ell_i}}{{\\rm d} t^{\\ell_i}} f(t)\n\t\\]\n\twhere in the inner sum $\\ell$ runs over all sequences $\\ell = (\\ell_1, \\ldots, \\ell_j)$, $\\ell_i \\in \\NN$, with\n\t$\\ell_1+\\ldots+\\ell_j = k$. Since the coefficients $C_\\ell$ are independent of $f$, by taking $f(t) = e^t$ and $k \\geq 2$,\n\twe get\n\t\\[\n\t\t0 = \\sum_{j = 1}^k j! (-1)^{j+1} \\sum_{\\ell} C_\\ell.\n\t\\]\n\tHence, for $f \\equiv \\phi$, and $k \\geq 2$, we obtain\n\t\\begin{align*}\n\t\t\\frac{{\\rm d}^k}{{\\rm d} t^k} \\log \\phi(t)\n\t\t&= \\sum_{j = 1}^k j! (-1)^{j+1} \\sum_{\\ell} C_\\ell \\prod_{i = 1}^j \\bigg(\\frac{\\phi^{(\\ell_i)}(t)}{\\phi(t)} - 1 + 1\\bigg)\n\t\t-\\sum_{j = 1}^k j! (-1)^{j+1} \\sum_{\\ell} C_\\ell \\\\\n\t\t&= \\sum_{j = 1}^k j! (-1)^{j+1} \\sum_{\\ell} C_\\ell \\sum_{\\stackrel{I \\subset \\{1, \\ldots, j\\}}{I \\neq \\emptyset}}\n\t\t\\prod_{i \\in I} \\bigg(\\frac{\\phi^{(\\ell_i)}(t)}{\\phi(t)} - 1\\bigg).\n\t\\end{align*}\n\tNow, the claim follows by \\eqref{lim:phi-1} and \\eqref{eq:11}.\n\\end{proof}\n\nIn view of \\eqref{eq:p_gauss}, \\eqref{eq:29}, and \\eqref{eq:30} our aim is to understand the critical points of the function\n\\begin{equation}\n\t\\label{psi}\n\t\\psi(u) = -\\frac{d}{2} \\log u -\\frac{L}{u} + \\log \\phi\\big(t(1-u)\\big), \\qquad u \\in (0, 1)\n\\end{equation}\nwhere\n\\begin{equation}\\label{A}\n\tL = \\frac{|y|^2}{4 t}.\n\\end{equation}\nWe note that\n\\begin{equation}\n\t\\label{eq:33}\n\t\\psi'(u) = -\\frac{d}{2} \\frac{1}{u} + \\frac{L}{u^2} - t \\frac{\\phi'\\big(t(1-u)\\big)}{\\phi\\big(t(1-u)\\big)},\n\\end{equation}\nand\n\\begin{equation}\n\t\\label{eq:37}\n\t\\psi''(u) = \\frac{d}{2} \\frac{1}{u^2} - 2 \\frac{L}{u^3} +\n\tt^2\\bigg\\{\n\t\\frac{\\phi''\\big(t(1-u)\\big)}{\\phi\\big(t(1-u)\\big)} -\n\t\\bigg(\\frac{\\phi'\\big(t(1-u)\\big)}{\\phi\\big(t(1-u)\\big)}\\bigg)^2\n\t\\bigg\\}.\n\\end{equation}\nThe following lemma is instrumental in treating the integral \\eqref{eq:p_gauss} restricted to the interval $(0, m)$.\n\\begin{lemma}\n\t\\label{lem:psi_prim}\n\tIf\n\t\\[\n\t\t\\frac{L}{t} \\geq m^2 + \\frac{C}{t}\n\t\\]\n\tfor certain $C > \\frac{d}{2} m$, then $\\psi'$ is decreasing and $\\psi'(m)>0$. Moreover, there is $T_0 > 0$ such that\n\tfor all $t \\geq T_0$,\n\t\\[\n\t\t\\int_0^m u^{-\\frac{d}{2}} e^{-\\frac{|y|^2}{4 tu}+\\log \\frac{\\Phi(t, u)}{t}} {\\: \\rm d} u\n\t\t\\leq m e^{\\psi(m)}.\n\t\\]\n\\end{lemma}\n\\begin{proof}\n\tIn view of \\eqref{eq:37} and Lemma \\ref{lem:1}, $\\psi''(u) < 0$ for all $u \\in (0, 1)$ provided that $L > d/4$.\n\tNext, by \\eqref{eq:33} we have\n\t\\[\n\t\t\\psi'(m) = -\\frac{d}{2} \\frac{1}{m}\n\t\t+ t\\bigg(1 - \\frac{\\phi'\\big(t(1-m)\\big)}{\\phi\\big(t(1-m)\\big)}\\bigg)\n\t\t+ \\frac{t}{m^2} \\bigg(\\frac{L}{t} - m^2\\bigg).\n\t\\]\n\tSince\n\t\\[\n\t\t-\\frac{d}{2} \\frac{1}{m}\n\t\t+ \\frac{t}{m^2} \\Big(\\frac{L}{t} - m^2\\Big)\n\t\t\\geq -\\frac{d}{2} \\frac{1}{m}+\\frac{C}{m^2}>0,\n\t\\]\n\tby \\eqref{lim:phi-1}, we get $\\psi'(m) > 0$ provided that $t$ is sufficiently large. Lastly, by \\eqref{eq:29} and\n\t\\eqref{psi} we get\n\t\\begin{align*}\n\t\t\\int_0^m u^{-\\frac{d}{2}} e^{-\\frac{|y|^2}{4 tu}+\\log \\frac{\\Phi(t, u)}{t}} {\\: \\rm d} u\n \t\t&\\leq \\int_0^m e^{\\psi(u)} {\\: \\rm d} u\n\t\\end{align*}\n\twhich is bounded by $m e^{\\psi(m)}$.\n\\end{proof}\n\nBefore we formulate the next result let us define\n\\[\n\t\\vphi(r) = \\int_0^\\infty e^{-s} \\phi(r s) {\\: \\rm d} s, \\qquad\\text{ for } r \\in [0, 1).\n\\]\nUsing the definition of the function $\\phi$ we can write\n\\begin{align*}\n\t\\vphi(r)\n\t&=\n\t\\sum_{j = 0}^\\infty \\frac{1}{j!} \\frac{r^j}{(m; m)_{j+1}} \\int_0^\\infty s^j e^{-s} {\\: \\rm d} s \\\\\n\t&=\n\t\\sum_{j = 0}^\\infty \\frac{r^j}{(m; m)_{j+1}}.\n\\end{align*}\n\n\\begin{theorem}\n\t\\label{thm:5}\n\tSuppose that $\\mathbf{Y}$ is Brownian motion in $\\RR^d$. Assume that $\\mathbf{X}$ is obtained from $\\mathbf{Y}$\n\tby partial resetting with factor $c \\in (0,1)$. Then for each $\\delta > 0$, the transition density of $\\mathbf{X}$\n\tsatisfies\n\t\\[\n\t\tp(t; 0, y)\n\t\t= e^{-t}\n\t\t(4\\pi t)^{-\\frac{d}{2}} e^{-\\frac{|y|^2}{4t}} \\bigg\\{\n\t\t1 + \\bigg(\\frac{4t^2}{|y|^2}\\bigg) \\vphi\\bigg(\\frac{4t^2}{\\norm{y}^2}\\bigg)+\n\t\t\\calO\\bigg(\\frac{t}{\\norm{y}^2}\\bigg)\n\t\t\\bigg\\}\n\t\\]\n\tas $t$ tends to infinity, uniformly in the region\n\t\\begin{equation}\n\t\t\\label{rg1}\n\t\t\\Big\\{(t, y) \\in \\RR_+ \\times \\RR^d : \\frac{|y|^2}{4t^2} \\geq 1 +\\delta \\Big\\}.\n\t\\end{equation}\n\\end{theorem}\n\\begin{proof}\n\tThank to \\eqref{eq:p_gauss}, we can write\n\t\\begin{equation}\n\t\t\\label{eq:58}\n\t\te^t (4 \\pi t)^{\\frac{d}{2}}\n\t\tp(t; 0, y) = e^{-\\frac{\\norm{y}^2}{4 t}} + t\n\t\t\\int_0^m u^{-\\frac{d}{2}}\n\t\tu^{-\\frac{d}{2}} e^{-\\frac{\\norm{y}^2}{4tu} + \\log \\frac{\\Phi(t, u)}{t}} {\\: \\rm d} u\n\t\t+t I\n\t\\end{equation}\n\twhere we have set\n\t\\[\n\t\tI = \\int_m^1 u^{-\\frac{d}{2}} e^{-\\frac{|y|^2}{4 tu}+\\log \\frac{\\Phi(t, u)}{t}} {\\: \\rm d} u.\n\t\\]\n\tUsing \\eqref{eq:30}, we get\n\t\\[\n\t\tI=\n\t\t\\int_m^1\n\t\tu^{-\\frac{d}{2}} e^{-\\frac{L}{u}}\n\t\t\\phi\\big(t(1-u)\\big) {\\: \\rm d} u.\n\t\\]\n\tLet us observe that for $(t, y)$ in the region \\eqref{rg1} we have $L \\geq (1+\\delta) t$.\n\tBy the change of variable $v = L(u^{-1}-1)$, we obtain\n\t\\begin{align*}\n\t\t\\int_m^1 u^{-\\frac{d}{2}} e^{-\\frac{L}{u}} \\phi\\big(t(1-u)\\big)\n\t\t{\\: \\rm d} u\n\t\t=\n\t\t\\frac{1}{L} e^{-L}\n\t\t\\int_0^{L (m^{-1}-1)} e^{-v}\n\t\t\\Big(1 + \\frac{v}{L} \\Big)^{\\frac{d}{2}-2}\n\t\t\\phi\\Big(\\frac{t}{L} \\frac{L v}{v+L} \\Big)\n\t\t{\\: \\rm d} v.\n\t\\end{align*}\n\tBy \\eqref{eq:10}, for $v > 0$ we have\n\t\\begin{align*}\n\t\te^{-v} \\phi\\Big(\\frac{t}{L} \\frac{L v}{v+L} \\Big)\n\t\t&\\leq\n\t\t\\frac{1}{(m; m)_\\infty}\n\t\t\\exp\\Big\\{-v + v \\frac{t}{L} \\frac{L}{v+L} \\Big\\} \\\\\n\t\t&\\leq\n\t\t\\frac{1}{(m; m)_\\infty}\n\t\t\\exp\\Big\\{-v + v \\frac{1}{1+\\delta} \\Big\\} \\\\\n\t\t&=\n\t\t\\frac{1}{(m; m)_\\infty} e^{-\\frac{\\delta}{1+\\delta} v}.\n\t\\end{align*}\n\tMoreover, for $v \\in [0, L(m^{-1} - 1) ]$,\n\t\\begin{align*}\n\t\t\\Big|\\Big(1 + \\frac{v}{L} \\Big)^{\\frac{d}{2}-2} - 1 \\Big|\n\t\t\\leq C \\frac{v}{L}.\n\t\\end{align*}\n\tHence,\n\t\\begin{align*}\n\t\t&\\bigg|\n\t\t\\int_0^{L(m^{-1}-1)}\n\t\te^{-v}\n\t\t\\Big(1 + \\frac{v}{L} \\Big)^{\\frac{d}{2}-2}\n\t\t\\phi\\Big(\\frac{t}{L} \\frac{L v}{v+L} \\Big)\n\t\t{\\: \\rm d} v\n\t\t-\n\t\t\\int_0^{L(m^{-1}-1)} e^{-v}\n\t\t\\phi\\Big(\\frac{t}{L} \\frac{L v}{v+L} \\Big)\n\t\t{\\: \\rm d} v\n\t\t\\bigg| \\\\\n\t\t&\\qquad\\qquad \\leq\n\t\t\\int_0^{L(m^{-1}-1)} e^{-v} \\phi\\Big(\\frac{t}{L} \\frac{L v}{v+L} \\Big)\n\t\t\\Big|\\Big(1 + \\frac{v}{L} \\Big)^{\\frac{d}{2}-2} - 1 \\Big|\n\t\t{\\: \\rm d} v \\\\\n\t\t&\\qquad\\qquad\\leq\n\t\t\\frac{C}{(m;m)_\\infty}\n\t\t\\frac1{L} \\int_0^\\infty e^{-\\frac{\\delta}{1+\\delta} v} v {\\: \\rm d} v.\n\t\\end{align*}\n\tSince $\\phi'$ is increasing, by the mean value theorem and \\eqref{eq:10},\n\t\\begin{align*}\n\t\t\\bigg|\n\t\t\\phi\\Big(\\frac{t}{L} \\frac{L v}{v+L} \\Big) - \\phi\\Big(\\frac{t}{L} v \\Big)\n\t\t\\bigg|\n\t\t&\\leq\n\t\t\\phi'\\Big(\\frac{t}{L} v \\Big) \\Big|\\frac{t}{L} v \\frac{L}{v+L} - \\frac{t}{L} v \\Big| \\\\\n\t\t&\\leq\n\t\t\\frac{v}{(m;m)_\\infty} \\exp\\Big\\{\\frac{t}{L} v\\Big\\} \\Big|\\frac{L}{v+L} - 1\\Big| \\\\\n\t\t&\\leq\n\t\t\\frac{v^2}{(m;m)_\\infty} \\frac1{L} \\exp\\Big\\{\\frac{t}{L} v\\Big\\} \\\\\n\t\t&\\leq\n\t\t\\frac1{(m;m)_\\infty} \\frac1{L} v^2 e^{\\frac{1}{1+\\delta} v}.\n\t\\end{align*}\n\tTherefore,\n\t\\begin{align*}\n\t\t&\\bigg|\n\t\t\\int_0^{L(m^{-1}-1)}\n\t\te^{-v}\n\t\t\\phi\\Big(\\frac{t}{L} \\frac{L v}{v+L} \\Big)\n\t\t{\\: \\rm d} v\n\t\t-\n\t\t\\int_0^{L(m^{-1}-1)} e^{-v} \\phi\\Big(\\frac{t}{L}v\\Big)\n\t\t{\\: \\rm d} v\n\t\t\\bigg| \\\\\n\t\t&\\qquad\\qquad\\leq\n\t\t\\int_0^{L(m^{-1}-1)}\n\t\te^{-v}\n\t\t\\bigg|\\phi\\Big(\\frac{t}{L} \\frac{L v}{v+L} \\Big) - \\phi\\Big(\\frac{t}{L}v\\Big) \\bigg|\n\t\t{\\: \\rm d} v \\\\\n\t\t&\\qquad\\qquad\\leq\n\t\t\\frac1{(m;m)_\\infty} \\frac1{L} \\int_0^\\infty e^{-\\frac{\\delta}{1+\\delta} v} v^2 {\\: \\rm d} v.\n\t\\end{align*}\n\tLastly, we can write\n\t\\begin{align*}\n\t\t\\int_0^{L(m^{-1}-1)} e^{-v} \\phi\\Big(\\frac{t}{L}v\\Big) {\\: \\rm d} v\n\t\t&=\n\t\t\\int_0^\\infty e^{-v} \\phi\\Big(\\frac{t}{L}v\\Big) {\\: \\rm d} v\n\t\t-\n\t\t\\int_{L(m^{-1}-1)}^\\infty\n\t\te^{-v} \\phi\\Big(\\frac{t}{L}v\\Big) {\\: \\rm d} v \\\\\n\t\t&=\n\t\t\\vphi\\Big(\\frac{t}{L}\\Big)\n\t\t-\n\t\t\\int_{L(m^{-1}-1)}^\\infty\n\t\te^{-v} \\phi\\Big(\\frac{t}{L}v\\Big) {\\: \\rm d} v\n\t\\end{align*}\n\tand so, by \\eqref{eq:10},\n\t\\begin{align*}\n\t\t\\int_{L(m^{-1}-1)}^\\infty\n\t\te^{-v} \\phi\\Big(\\frac{t}{L}v\\Big) {\\: \\rm d} v\n\t\t&\\leq\n\t\t\\frac1{(m;m)_\\infty}\n\t\t\\int_{L(m^{-1}-1)}^\\infty \\exp\\Big\\{-\\Big(1-\\frac{t}{L}\\Big) v\\Big\\} {\\: \\rm d} v \\\\\n\t\t&\\leq\n\t\t\\frac1{(m;m)_\\infty}\n\t\t\\int_{L(m^{-1}-1)}^\\infty e^{-\\frac{\\delta}{1+\\delta} v} {\\: \\rm d} v \\\\\n\t\t&\\leq\n\t\t \\frac1{(m;m)_\\infty} e^{-L (1-m) \\frac{\\delta}{1+\\delta}}\n\t\t \\int_0^\\infty e^{-(1-m) \\frac{\\delta}{1+\\delta} v} {\\: \\rm d} v.\n\t\\end{align*}\n\tSummarizing, we obtain\n\t\\begin{align*}\n\t\tI=\n\t\te^{-L} \\Big( \\frac{t}{L}\\Big)\n\t\t\\Big\\{ \\vphi\\Big(\\frac{t}{L}\\Big) + \\calO\\big(L^{-1}\\big)\\Big\\}.\n\t\\end{align*}\n\tHence, by \\eqref{eq:58} and Lemma \\ref{lem:psi_prim}\n\t\\begin{align*}\n\t\t\\Big|\n\t\te^t (4 \\pi t)^{\\frac{d}{2}} p(t; 0, y) - e^{-L} - t I\n\t\t\\Big|\n\t\t&\\leq\n\t\tt \\int_0^m u^{-\\frac{d}{2}} e^{-\\frac{L}{u} + \\log \\frac{\\Phi(t, u)}{t}} {\\: \\rm d} u \\\\\n\t\t&\\leq\n\t\tm t e^{\\psi(m)},\n\t\\end{align*}\n\tprovided that $t$ is sufficiently large. To complete the proof we need to show that\n\t\\[\n\t\tt e^{\\psi(m)+L}=\\calO\\big(L^{-1}\\big).\n\t\\]\n\tTo see this, we apply \\eqref{eq:10} to get\n\t\\begin{align*}\n\t\tt e^{\\psi(m)+L}\n\t\t&\\leq C_1 \\exp\\Big\\{-\\frac{L}{m} +t(1-m)+L\\Big\\}\\\\\n\t\t&=C_1 \\exp\\Big\\{ -L (1-m) \\Big(\\frac{1}{m} - \\frac{t}{L}\\Big) \\Big\\}\\\\\n\t\t&\\leq C_1 \\exp\\Big\\{ -L (1-m) \\Big(\\frac{1}{m} -1\\Big) \\Big\\}.\n\t\\end{align*}\n\tand the theorem follows.\n\\end{proof}\n\n\\begin{corollary}\n\tUnder the assumptions of Theorem \\ref{thm:5},\n\t\\[\n\t\t\\lim_{t \\to +\\infty}\n\t\t\\frac{p(t;0,y)}{\\abs{y}^{-\\frac{d-1}{2}} e^{-\\abs{y}}} = 0\n\t\\]\n\tuniformly in the region \\eqref{rg1}.\n\\end{corollary}\n\\begin{proof}\n\tSince $2\\sqrt{L t} = |y|$, by Theorem~\\ref{thm:5} we get\n\t\\[\n\t\t\\lim_{t \\to +\\infty}\n\t\t\\frac{p(t;0,y)}{\\abs{y}^{-\\frac{d-1}{2}} e^{-\\abs{y}}}\n\t\t=\n\t\t\\lim_{t \\to +\\infty}\n\t\tt^{-\\frac12} \\exp\\Big\\{-t-L -\\frac{d-1}{2}\\log t +2\\sqrt{L t} +\\frac{d-1}{2}\\log |y|\\Big\\}.\n\t\\]\n \tNow, it is sufficient to observe that in the region \\eqref{rg1},\n\t\\[\n\t\t-t \\Big(\\sqrt{\\frac{L}{t}}-1\\Big)^2 +\\frac{d}{2} \\log \\sqrt{\\frac{L}{t}}\n\t\\]\n\tis uniformly bounded from above.\n\\end{proof}\n\n\\begin{theorem}\n\t\\label{thm:6}\n\tSuppose that $\\mathbf{Y}$ is a Brownian motion in $\\RR^d$. Assume that $\\mathbf{X}$ is obtained from $\\mathbf{Y}$\n\tby partial resetting with factor $c\\in(0,1)$. Then for each $\\delta > 0$, the transition density of $\\mathbf{X}$\n\tsatisfies\n\t\\[\n\t\tp(t; 0, y)\n\t\t=\n\t\t\\frac{1}{2} \\frac1{(m;m)_\\infty} (2\\pi)^{-\\frac{d-1}{2}} \\abs{y}^{-\\frac{d-1}{2}} e^{-\\abs{y}}\n\t\t\\Big(1 + \\calO\\big(t^{-1}\\big)\\Big)\n\t\\]\n\tas $t$ tends to infinity, uniformly in the region\n\t\\begin{equation}\t\n\t\t\\label{eq:38}\n\t\t\\Big\\{(t, y) \\in \\RR_+ \\times \\RR^d : m^2 +\\delta \\leq \\frac{\\norm{y}^2}{4t^2} \\leq 1 - \\delta \\Big\\}.\n\t\\end{equation}\n\\end{theorem}\n\\begin{proof}\n\tNotice that for $(t, y)$ in the region \\eqref{eq:38}, $L$ is comparable to $t$. Now, let us observe that\n\t\\[\n\t\t\\psi'(1) = -\\frac{d}{2} + t \\bigg(\\frac{L}{t} - \\frac{\\phi'(0)}{\\phi(0)}\\bigg),\n\t\t\\quad\\text{ and }\\quad\n\t\t\\frac{\\phi'(0)}{\\phi(0)}=\\frac1{1-m^2},\n\t\\]\n\tthus $\\psi'(1)<0$. Using now Lemma~\\ref{lem:psi_prim} we conclude that $\\psi$ has the unique critical point $u_0$. Moreover,\n\t$u_0$ belongs to $(m, 1)$. In fact, we claim that $u_0 \\in (m, 1-\\delta')$ for certain $\\delta' > 0$. Indeed, suppose that\n\t$u_0$ tends to $1$. We can assume that $t(1-u_0)$ converges to $g \\in [0, \\infty]$. Since $u_0$ satisfies \\eqref{eq:33},\n\twe have\n\t\\[\n\t\t\\frac{1}{u_0^2} - 1 = \\frac{d}{2 L} \\frac{1}{u_0} +\n\t\t\\frac{t}{L} \\bigg( \\frac{\\phi'\\big(t(1-u_0) \\big)}{\\phi\\big(t(1-u_0) \\big)}\n\t\t-\n\t\t\\frac{\\phi'(g)}{\\phi(g)}\\bigg) + \\frac{t}{L}\\bigg(\\frac{\\phi'(g)}{\\phi(g)} - \\frac{L}{t}\\bigg).\n\t\\]\n\tHence\n\t\\[\n\t\t\\frac{t}{L}\\bigg(\\frac{\\phi'(g)}{\\phi(g)} - \\frac{L}{t}\\bigg) = o(1)\n\t\\]\n\twhich leads to a contradiction because according to \\eqref{eq:11},\n\t\\[\n\t\t1 \\leq \\frac{\\phi'(t)}{\\phi(t)} \\leq \\frac{1}{1-m^2}\n\t\\]\n\tfor all $t \\geq 0$. This proves the claim.\n\n\tNext, we determine the asymptotic behavior of $u_0$. Since $u_0$ solves \\eqref{eq:33}, we obtain\n\t\\[\n\t\tu_0 = \\frac{2L}{\\dfrac{d}{2} +\n\t\t\\sqrt{\\Big(\\dfrac{d}{2}\\Big)^2 + 4 L t \\dfrac{\\phi'(t(1-u_0))}{\\phi(t(1-u_0))}}}.\n\t\\]\n\tBecause $t(1-u_0)$ tends to infinity, by \\eqref{lim:phi-1},\n\t\\[\n\t\t\\frac{\\phi'\\big(t(1-u_0) \\big)}{\\phi\\big(t(1-u_0)\\big)}\n\t\t=\n\t\t1 + o\\big(t^{-2}\\big),\n\t\\]\n\tand consequently\n\t\\begin{align}\n\t\t\\nonumber\n\t\t\\frac{1}{u_0}\n\t\t&=\n\t\t\\frac{d}{4 L}\n\t\t+\\sqrt{\\Big(\\frac{d}{2}\\Big)^2 \\frac{1}{4 L^2} + \\frac{t}{L} + \\frac{t}{L}\n\t\t\\Big(\\frac{\\phi'(t(1-u_0))}{\\phi(t(1-u_0))} - 1\\Big)} \\\\\n\t\t\\nonumber\n\t\t&=\n\t\t\\frac{d}{4 L}\n\t\t+\\sqrt{\\frac{t}{L}}\n\t\t\\sqrt{\n\t\t1+ \\Big(\\frac{d}{2}\\Big)^2 \\frac{1}{4L t} + \\Big(\\frac{\\phi'(t(1-u_0))}{\\phi(t(1-u_0))} - 1\\Big)} \\\\\n\t\t\\nonumber\n\t\t&=\n\t \t\\frac{d}{4 L}+\\sqrt{\\frac{t}{L}}\n\t\t\\Big(1 + \\calO\\big(t^{-2}\\big)\\Big) \\\\\n\t\t\\label{eq:48}\n\t\t&= \\frac{d}{4 L} + \\sqrt{\\frac{t}{L}} + \\calO\\big(t^{-2}\\big).\n\t\\end{align}\n\tHence\n\t\\begin{align*}\n\t\tu_0\n\t\t&= \\frac{1}{ \\frac{d}{4L } + \\sqrt{\\frac{t}{L}} + \\calO\\big(t^{-2}\\big)} \\\\\n\t\t&= \\sqrt{\\frac{L}{t}} \\frac{1}{1+\\frac{d}{4 \\sqrt{t L}} + \\calO\\big(t^{-2}\\big)} \\\\\n\t\t&= \\sqrt{\\frac{L}{t}} \\Big(1 - \\frac{d}{4 \\sqrt{t L}} + \\calO\\big(t^{-2}\\big)\\Big),\n\t\\end{align*}\n\tand so\n\t\\begin{equation}\n\t\t\\label{eq:47}\n\t\tu_0 = \\sqrt{\\frac{L}{t}} - \\frac{d}{4 t} + \\calO\\big(t^{-2}\\big).\n\t\\end{equation}\n\tWe now establish the asymptotic behavior of $\\psi(u_0)$ and $\\psi''(u_0)$. By \\eqref{lim:phi-2},\n\t\\[\n\t\t\\frac{\\phi\\big(t(1-u_0)\\big)}{e^{t(1-u_0)}} = \\frac1{(m;m)_\\infty} + o\\big(t^{-1}\\big),\n\t\\]\n\tand\n\t\\[\n\t\t\\log \\phi\\big(t(1-u_0)\\big) = \\log \\frac1{(m;m)_\\infty} + t(1-u_0) + o\\big(t^{-1}\\big).\n\t\\]\n\tBy \\eqref{eq:48} and \\eqref{eq:47} we obtain\n\t\\begin{align*}\n\t\t\\frac{L}{u_0} &= \\frac{d}{4} + \\sqrt{L t} + \\calO\\big(t^{-1}\\big), \\\\\n\t\tt u_0 &= \\sqrt{L t} - \\frac{d}{4} + \\calO\\big(t^{-1}\\big), \\\\\n\t\t\\log u_0 &=\\log \\sqrt{\\frac{L}{t}}+\\calO\\big(t^{-1}\\big).\n\t\\end{align*}\n\tTherefore,\n\t\\[\n\t\t\\log \\phi\\big(t(1-u_0)\\big) = \\log \\frac1{(m;m)_\\infty}+ t - \\sqrt{L t} + \\frac{d}{4} + \\calO\\big(t^{-1}\\big),\n\t\\]\n\tand\n\t\\begin{align}\n\t\t\\nonumber\n\t\t\\psi(u_0)\n\t\t&= -\\frac{d}{2} \\log u_0 - \\frac{L}{u_0} + \\log \\phi\\big(t(1-u_0)\\big) \\\\\n\t\t\\label{eq:39}\n\t\t&= \\log \\frac1{(m;m)_\\infty} -\\frac{d}{2} \\log \\sqrt{\\frac{L}{t}} - 2 \\sqrt{L t} + t + \\calO\\big(t^{-1}\\big).\n\t\\end{align}\n\tUsing \\eqref{eq:48} and \\eqref{lim:phi-1} supported by \\eqref{eq:11} we can write\n\t\\begin{align}\n\t\t\\psi''(u_0)\n\t\t&= \\frac{d}{2} \\frac{1}{u_0^2} - 2 \\frac{L}{u_0^3} + t^2\n\t\t\\bigg\\{\\frac{\\phi''\\big(t(1-u_0)\\big)}{\\phi\\big(t(1-u_0)\\big)} -\n\t\t\\bigg(\\frac{\\phi'\\big(t(1-u_0)\\big)}{\\phi\\big(t(1-u_0)\\big)} \\bigg)^2\\bigg\\} \\nonumber \\\\\n\t\t\\nonumber\n\t\t&= \\frac{d}{2} \\frac{t}{L} - 2 t \\sqrt{\\frac{t}{L}} + \\calO\\big(t^{-1}\\big) \\\\\n\t\t&= -2t \\sqrt{\\frac{t}{L}} \\Big( 1 + \\calO\\big(t^{-1}\\big)\\Big).\\label{eq:psi_bis_asymp}\n\t\\end{align}\n\tWe are now ready to study the asymptotic behavior of the transition density. By \\eqref{eq:p_gauss},\n\t\\eqref{eq:30} and \\eqref{psi}, we can write\n\t\\begin{equation}\n\t\t\\label{eq:61}\n\t\te^t (4 \\pi t)^{\\frac{d}{2}} p(t; 0, y)\n\t\t=\n\t\te^{-L} + t \\int_0^m u^{-\\frac{d}{2}} e^{-\\frac{|y|^2}{4 tu}+\\log \\frac{\\Phi(t, u)}{t}} {\\: \\rm d} u\n\t\t+ t I\n\t\\end{equation}\n\twhere\n\t\\[\n\t\tI = \\int_m^1 e^{\\psi(u)} {\\: \\rm d} u.\n\t\\]\n\tSince\n\t\\[\n\t\tm^2 + \\delta \\leq \\frac{L}{t} \\leq 1 - \\delta,\n\t\\]\n\tand $L > d/2$, by Lemma \\ref{lem:1} for all $u \\in (m, 1)$,\n\t\\begin{align}\n\t\t\\label{eq:53}\n\t\t\\psi''(u)\n\t\t&< \\frac{d}{2} \\frac{1}{u^2} - \\frac{L}{u^2}- L\\leq - L\n\t\t\\leq -m^2 t.\n\t\\end{align}\n\tOur aim is to find the asymptotic behavior of $I$. Let us first focus on the integral over\n\t$(m, u_0 - \\eta) \\cup (u_0 + \\eta, 1)$, for certain $\\eta > 0$. By the Taylor's theorem and \\eqref{eq:53}, for\n\t$u \\in (m, 1)$ we get\n\t\\begin{align*}\n\t\t\\psi(u)\n\t\t&= \\psi(u_0) + \\frac{1}{2} \\psi''\\big(\\lambda u + (1-\\lambda) u_0 \\big) (u - u_0)^2 \\\\\n\t\t&\\leq \\psi(u_0) - \\frac{m^2}{2} t (u-u_0)^2.\n\t\\end{align*}\n\tTherefore, using \\eqref{eq:psi_bis_asymp},\n\t\\begin{align}\n\t\t\\nonumber\n\t\t\\bigg(\\int_m^{u_0-\\eta } + \\int_{u_0+\\eta}^1 \\bigg) e^{\\psi(u)} {\\: \\rm d} u\n\t\t&\\leq e^{\\psi(u_0)}\n\t\t\\bigg(\\int_m^{u_0-\\eta } + \\int_{u_0+\\eta}^1 \\bigg) e^{-\\frac{m^2}{2}t (u-u_0)^2} {\\: \\rm d} u \\\\\n\t\t\\nonumber\n\t\t& \\leq\n\t\te^{\\psi(u_0)} e^{-\\frac{m^2 \\eta^2}{2} t}\\\\\n\t\t&\\leq C t^{-1} e^{\\psi(u_0)} (-\\psi''(u_0))^{-\\frac12}.\n\t\t\\label{eq:51}\n\t\\end{align}\n\tNext, we tread the integral over $(u_0-\\eta,u_0+\\eta)$. We write\n\t\\begin{equation}\n\t\t\\label{eq:45}\n\t\t\\int_{\\abs{u-u_0} \\leq \\eta} e^{\\psi(u)} {\\: \\rm d} u\n\t\t=\n\t\te^{\\psi(u_0)} \\int_{\\abs{u - u_0} \\leq \\eta} e^{\\frac{1}{2} \\psi''(u_0) (u-u_0)^2} e^{\\theta(u)} {\\: \\rm d} u\n\t\\end{equation}\n\twhere\n\t\\[\n\t\t\\theta(u) = \\psi(u) - \\psi(u_0) - \\frac{1}{2} \\psi''(u_0) (u-u_0)^2.\n\t\\]\n\tFor $k \\geq 2$, we compute\n\t\\[\n\t\t\\psi^{(k)}(u) = (-1)^{k} k! \\frac{d}{2} \\frac{1}{u^k} + (-1)^{k+1} k! \\frac{L}{u^{k+1}}\n\t\t+ (-1)^k t^k (\\log \\phi)^{(k)}\\big(t(1-u)\\big).\n\t\\]\n\tHence, by Proposition~\\ref{prop:log_phi}, there is $C_k$ such that for all $u \\in (m, u_0+\\eta)$ and $t \\geq 1$,\n\t\\begin{equation}\n\t\t\\label{eq:41}\n\t\t\\big| \\psi^{(k)}(u) \\big| \\leq C_k t,\n\t\\end{equation}\n\tthus by the Taylor's theorem\n\t\\begin{align}\n\t\t\\label{eq:54}\n\t\t\\abs{\\theta(u)}\n\t\t\\leq \\frac{C_3}{3!} t \\abs{u-u_0}^3.\n\t\\end{align}\n\tIf $\\abs{u - u_0} \\leq \\eta$, by \\eqref{eq:53}, we obtain\n\t\\begin{equation}\n\t\t\\label{eq:46}\n\t\t\\abs{\\theta(u)} \\leq -\\frac{1}{4} \\psi''(u_0) \\abs{u - u_0}^2\n\t\\end{equation}\n\tprovided that\n\t\\[\n\t\t\\eta \\leq m^2 \\frac{3}{2 C_3}.\n\t\\]\n\tAnother application of the Taylor's theorem together with \\eqref{eq:41} gives\n\t\\begin{equation}\n\t\t\\label{eq:44}\n\t\t\\Big|\n\t\t\\theta(u) - \\frac{1}{3!} \\psi'''(u_0) (u-u_0)^3\n\t\t\\Big|\n\t\t\\leq\n\t\t\\frac{C_4}{4!} t \\abs{u-u_0}^4.\n\t\\end{equation}\n\tNow, we write\n\t\\begin{align*}\n\t\te^{\\theta(u)}\n\t\t&= \\Big(e^{\\theta(u)} - 1 - \\theta(u)\\Big) \\\\\n\t\t&\\phantom{=}+\n\t\t\\Big(\\theta(u) - \\frac{1}{3!} \\psi'''(u_0)(u-u_0)^3\\Big) \\\\\n\t\t&\\phantom{=}+\n\t\t\\frac{1}{3!} \\psi'''(u_0)(u-u_0)^3 + 1,\n\t\\end{align*}\n\tand we split the integral \\eqref{eq:45} into four corresponding integrals. Observe that for $x \\in \\RR$,\n\t\\begin{equation}\n\t\t\\label{eq:49}\n\t\t\\big| e^x - 1 - x\\big| \\leq \\frac{1}{2} x^2 e^{|x|}.\n\t\\end{equation}\n\tHence, by \\eqref{eq:54} and \\eqref{eq:46}, the first integrand can be bounded as follows\n\t\\begin{align*}\n\t\t\\Big|e^{\\theta(u)} - 1 - \\theta(u) \\Big|\n\t\t\\leq\n\t\tC t^2 \\abs{u-u_0}^6 e^{-\\frac{1}{4} \\psi''(u_0)(u-u_0)^2}.\n\t\\end{align*}\n\tTherefore, by \\eqref{eq:53},\n\t\\begin{align*}\n\t\t&\\bigg|\n\t\t\\int_{\\abs{u-u_0} \\leq \\eta}\n\t\te^{\\frac{1}{2}\\psi''(u_0) (u-u_0)^2} \\big(e^{\\theta(u)} - 1 - \\theta(u) \\big) {\\: \\rm d} u\n\t\t\\bigg| \\\\\n\t\t&\\qquad\\qquad\\leq\n\t\tC t^2 \\int_{\\abs{u - u_0} \\leq \\eta} e^{\\frac{1}{4} \\psi''(u_0)(u-u_0)^2} \\abs{u-u_0}^6 {\\: \\rm d} u \\\\\n\t\t&\\qquad\\qquad =\n\t\tC t^2 (-\\psi''(u_0))^{-3-\\frac{1}{2}}\n\t\t\\int_{\\abs{u} \\leq \\eta \\sqrt{-\\psi''(u_0)}} e^{-\\frac{1}{4} \\abs{u}^2} \\abs{u}^6 {\\: \\rm d} u\n\t\t\\\\\n\t\t&\\qquad\\qquad\\leq C t^{-1} (-\\psi''(u_0))^{-\\frac{1}{2}}.\n\t\\end{align*}\n\tFor the second integral, by \\eqref{eq:44} and \\eqref{eq:53} we obtain\n\t\\begin{align*}\n\t\t&\\bigg|\n\t\t\\int_{\\abs{u - u_0} \\leq \\eta} e^{\\frac{1}{2}\\psi''(u_0) (u-u_0)^2}\n\t\t\\Big(\\theta(u) - \\frac{1}{3!} \\psi'''(u_0)(u-u_0)^3 \\Big) {\\: \\rm d} u\n\t\t\\bigg|\\\\\n\t\t&\\qquad\\qquad\\leq\n\t\tC t \\int_{\\abs{u-u_0} \\leq \\eta} e^{\\frac{1}{2}\\psi''(u_0) (u-u_0)^2} \\abs{u-u_0}^4 {\\: \\rm d} u\\\\\n\t\t&\\qquad\\qquad\\leq\n\t\tC t (-\\psi''(u_0))^{-2 - \\frac{1}{2}} \\int_{\\abs{u} \\leq \\eta \\sqrt{-\\psi''(u_0)}}\n\t\te^{-\\frac{1}{2} \\abs{u}^2} \\abs{u}^4 {\\: \\rm d} u \\\\\n\t\t&\\qquad\\qquad\\leq\n\t\tC t^{-1} (\\psi''(u_0))^{-\\frac{1}{2}}.\n\t\\end{align*}\n\tThe third integral equals zero. Lastly, by \\eqref{eq:53}, we have\n\t\\begin{align*}\n\t\t\\int_{\\abs{u-u_0} \\leq \\eta}e^{\\frac{1}{2}\\psi''(u_0) (u-u_0)^2} {\\: \\rm d}u\n\t\t&=(-\\psi''(u_0))^{-\\frac{1}{2}}\n\t\t\\int_{\\abs{u} \\leq \\eta \\sqrt{-\\psi''(u_0)}} e^{-\\frac{1}{2} \\abs{u}^2} {\\: \\rm d} u\\\\\n\t\t&=(-\\psi''(u_0))^{-\\frac{1}{2}} \\left(\\sqrt{2\\pi} + \\int_{\\abs{u}\n\t\t\\geq \\eta \\sqrt{-\\psi''(u_0)}} e^{-\\frac{1}{2} \\abs{u}^2} {\\: \\rm d} u \\right)\\\\\n\t\t&= (-\\psi''(u_0))^{-\\frac{1}{2}} \\sqrt{2\\pi} \\Big(1 + \\calO(t^{-1})\\Big).\n\t\\end{align*}\n\tSummarizing, we get\n\t\\begin{align*}\n\t\tI = e^{\\psi(u_0)} (-\\psi''(u_0))^{-\\frac{1}{2}} \\sqrt{2\\pi} \\Big(1 + \\calO(t^{-1})\\Big).\n\t\\end{align*}\n\tHence, by \\eqref{eq:39} and \\eqref{eq:psi_bis_asymp},\n\t\\begin{align*}\n\t\tI\n\t\t&=\n\t\t\\frac1{(m;m)_\\infty} \\Big(\\frac{L}{t}\\Big)^{-\\frac{d}{4}}\n\t\te^{- 2 \\sqrt{L t} + t} \\Big( 2t \\sqrt{\\frac{t}{L}} \\Big)^{-\\frac12} \\sqrt{2\\pi}\n\t\t\\Big(1 + \\calO(t^{-1})\\Big)\\\\\n\t\t&= \\frac1{(m;m)_\\infty} (2 \\pi)^{-\\frac{d-1}{2}} |y|^{-\\frac{d-1}{2}}e^{-|y|} \\Big(1 + \\calO(t^{-1})\\Big).\n\t\\end{align*}\n\tMoreover, by \\eqref{eq:61} and Lemma \\ref{lem:1} we get\n\t\\begin{align*}\n\t\t\\Big|\n\t\tp(t; 0, y) - e^{-t} (4 \\pi t)^{-\\frac{d}{2}}e^{-L} -\n\t\tt e^{-t} (4 \\pi t)^{-\\frac{d}{2}} I\n\t\t\\Big|\n\t\t\\leq\n\t\tm t^{1-\\frac{d}{2}} e^{-t+\\psi(m)}.\n\t\\end{align*}\n\tTherefore, to finish the proof we need to show that\n\t\\begin{equation}\n\t\t\\label{eq:62}\n\t\tt^{-\\frac{d}{2}}e^{-t-L}\n\t\t=\n\t\t\\abs{y}^{-\\frac{d-1}{2}} e^{-\\abs{y}} \\calO(t^{-1}),\n\t\\end{equation}\n\tand\n\t\\begin{equation}\n\t\t\\label{eq:63}\n\t\tt^{1-\\frac{d}{2}}e^{-t+\\psi(m)}\n\t\t=\n\t\t\\abs{y}^{-\\frac{d-1}{2}} e^{-\\abs{y}} \\calO(t^{-1}).\n\t\\end{equation}\n\tNotice that $t^{-\\frac{d}{2}}$ is comparable to $t^{-\\frac12}|y|^{-\\frac{d-1}{2}}$. Since $L \\leq t(1-\\delta)$,\n\twe have\n\t\\[\n\t\t\\Big(\\sqrt{\\frac{L}{t}} - 1 \\Big)^2 \\geq (1-\\sqrt{1-\\delta})^2\n\t\\]\n\twhich implies that\n\t\\[\n\t\t-t - L \\leq -\\norm{y}- (1-\\sqrt{1-\\delta})^2 t\n\t\\]\n\tproving \\eqref{eq:62}. Since $t^{1-\\frac{d}{2}}$ is comparable to $t^{\\frac12}|y|^{-\\frac{d-1}{2}}$, and\n\t$L \\geq t(m^2 + \\delta)$, we get\n\t\\[\n\t\t\\Big(\\sqrt{\\frac{L}{t}} - m \\Big)^2 \\geq (\\sqrt{m^2+\\delta}-m)^2=\\delta^*\n\t\\]\n\twhich implies that\n\t\\[\n\t\t- \\frac{L}{m}- m t \\leq -\\norm{y}- \\frac{(\\sqrt{m^2+\\delta}-m)^2}{m} t.\n\t\\]\n\tIn view of \\eqref{psi} and \\eqref{eq:10}, we have\n\t\\[\n\t\t-t+\\psi(m)\\leq -\\frac{L}{m} -mt +C,\n\t\\]\n\twhich proves \\eqref{eq:63}, and the theorem follows.\n\\end{proof}\n\n\\begin{remark}\n\tWe note that the asymptotic behavior of $p(t; 0, y)$ in Theorem~\\ref{thm:6} remain true if $m^2+\\delta \\leq L/t$\n\tis replaced by\n\t\\[\n\t\t\\Big(m+\\sqrt{\\frac{\\log f(t)}{t}} \\Big)^2\n\t\t\\leq \\frac{L}{t}\n\t\\]\n\twhere $f$ is a fixed positive function such that\n\t\\[\n\t\t\\liminf_{t\\to\\infty} t^{-\\frac{3m}{2}} f(t) >0.\n\t\\]\n\tIndeed, Lemma~\\ref{lem:psi_prim} still applies, and the only change required in the proof is in showing that\n\t\\[\n\t\tt^{\\frac12} e^{-t+\\psi(m)}=e^{-|y|}\\calO(t^{-1}),\n\t\\]\n\tor equivalently\n\t\\[\n\t\tt^{\\frac12} \\exp\\Big\\{-\\frac{t}{m}\\Big( \\sqrt{\\frac{L}{t}} -m\\Big)^2\\Big\\} =\\calO(t^{-1}),\n\t\\]\n\twhich is guaranteed by the proposed condition.\n\\end{remark}\n\n\\begin{remark}\n\t\\label{nessgeneralgaussian}\n\tTheorems~\\ref{thm:5} and~\\ref{thm:6} can be easily generalized to the case of\n\t$Y_t=\\Upsilon B_t$ for $\\Upsilon \\in \\GL(\\mathbb{R}, d)$. In such a case, the counterpart of \\eqref{eq:p_gauss} is given by\n\t\\[\n\t\tp(t; 0, y) = e^{-t} (\\det \\Upsilon)^{-1} (2\\pi t)^{-\\frac{d}{2}} e^{-\\frac{|\\Upsilon^{-1} y|^2}{4t}}\n\t\t+t e^{-t}(4\\pi t)^{-\\frac{d}{2}} \\int_0^1 u^{-\\frac{d}{2}} e^{-\\frac{|\\Upsilon^{-1} y|^2}{4 tu}+\n\t\t\\log \\frac{\\Phi(t, u)}{t}} {\\: \\rm d} u.\n\t\\]\n\tThen $\\eqref{psi}$ takes the form\n\t\\[\n\t\t\\psi(u) = -\\frac{d}{2} \\log u -\\frac{L}{u} + \\log \\phi\\big(t(1-u)\\big)\n\t\\]\n\twhere\n\t\\[\n\t\tL = \\frac{|\\Upsilon^{-1}y|^2}{2 t},\n\t\\]\n\tcf. \\eqref{A}. The statements of Theorems~\\ref{thm:5} and~\\ref{thm:6} remain true if we replace $\\frac{|y|^2}{4 t}$\n\tby $\\frac{|\\Upsilon^{-1}y|^2}{2 t}$. All the proofs are the same.\n\\end{remark}\n\n\\appendix\n\n\\section{Strictly stable processes}\n\\label{appendix:A}\nWe begin with a general observation on L{\\'e}vy process in $\\mathbb{R}^d$. By \\cite[Theorem 8.1]{MR1739520} there is\na one-to-one correspondence between L{\\'e}vy processes and generating triplets $(A, \\nu, \\gamma)$ where $A$ is nonnegative\ndefinite real $d \\times d$ matrix, $\\nu$ a Borel measure on $\\RR^d$ such that\n\\[\n\t\\int_{\\RR^d} \\big(1 \\land \\norm{x}^2\\big) \\nu({\\rm d} x) < \\infty,\n\\]\nand $\\gamma \\in \\RR^d$. The triplet are helpful in identification of the generator of the process, see\n\\cite[Theorem 31.5]{MR1739520}, which for sufficiently smooth functions $f$ can be given by the following expression\n\\begin{align*}\n\t\\mathscr{L} f(x)\n\t&= \\sum_{j, k = 1}^d a_{jk} \\partial_{x_j} \\partial_{x_k} f(x) + \\sprod{\\nabla f(x)}{\\gamma}\n\t+\n\t\\int_{\\RR^d} \\Big(f(x+z) - f(x) - \\ind{\\{\\abs{z} < 1\\}} \\sprod{\\nabla f(x)}{z}\\Big) \\nu({\\rm d} z).\n\\end{align*}\nWe are going to describe the infinitesimal generator of the semigroup associated with the process considered\non $\\calC_0(\\RR^d)$ as well as $L^r(\\RR^d)$, $r \\in [1, \\infty)$. Let us recall that\n\\[\n\t\\mathcal{B}_r =\n\t\\begin{cases}\n\t\tL^r(\\RR^d) &\\text{if } r \\in [1, \\infty), \\\\\n\t\t\\calC_0(\\RR^d) &\\text{if } r = \\infty.\n\t\\end{cases}\n\\]\n\\begin{proposition}\n\t\\label{prop:9}\n\tLet $\\mathbf{Y}$ be a L{\\'e}vy process in $\\mathbb{R}^d$ and $r \\in [1, \\infty]$. Then the family of operators defined\n\ton $\\mathcal{B}_r$ as\n\t\\[\n\t\t\\Big(f \\mapsto \\EE[f(Y_t + \\cdot)]: t > 0\\Big)\n\t\\]\n\tforms a strongly continuous contraction semigroup. Let $\\mathscr{L}_r$ be its infinitesimal generator with domain\n\t$D(\\mathscr{L}_r)$. Moreover, if $f \\in \\calC_0^2(\\RR^d)$ is such that $\\partial_x^{\\mathbf{a}} f \\in \\mathcal{B}_r$\n\tfor each $\\mathbf{a} \\in \\NN_0^d$, $\\abs{\\mathbf{a}} \\leq 2$, then $f \\in D(\\mathscr{L}_r)$ and\n\t\\[\n\t\t\\mathscr{L}_r f = \\mathscr{L} f.\n\t\\]\n\\end{proposition}\n\\begin{proof}\n\tIt is well-known that the contractivity and strong continuity are consequences of Minkowski's integral inequality and\n\tthe continuity of the shift operator, while the semigroup property follows by the Markov property of the process.\n\tSince for $f \\in \\calC_0^2(\\RR^d)$,\n\t\\[\n\t\t\\big\\|\\mathscr{L}f \\big\\|_{\\mathcal{B}_r} \\leq c\n\t\t\\sum_{|\\mathbf{a}|\\leq 2} \\left\\|\\partial^{\\mathbf{a}} f \\right \\|_{ \\mathcal{B}_r}\n\t\\]\n\twe have $\\mathscr{L}f \\in \\mathcal{B}_r$, because $\\partial^{\\mathbf{a}} f \\in \\mathcal{B}_r$ for each\n\t$\\mathbf{a} \\in \\NN_0^d$, $\\abs{\\mathbf{a}} \\leq 2$. Next, by \\cite[Theorem 31.5]{MR1739520}, we get\n\t\\[\n\t\t\\bigg\\|\n\t\t\\frac{\\EE\\big[f(Y_t + x)\\big]-f(x)}{t} - \\mathscr{L} f(x)\n\t\t\\bigg\\|_{\\mathcal{B}_r(x)}\n\t\t\\leq\n\t\t\\frac1{t} \\int_0^t\n\t\t\\Big\\|\n\t\t\\EE\\big[\\mathscr{L}f(Y_s + x)- \\mathscr{L} f(x) \\big]\n\t\t\\Big\\|_{\\mathcal{B}_r(x)} {\\rm d}s.\n\t\\]\n\tBy the strong continuity in $\\mathcal{B}_r(x)$, the right hand-side in the last inequality\n\tconverges to zero as $t$ tends to zero.\n\\end{proof}\n\nL{\\'e}vy processes in $\\RR^d$ are in one-to-one correspondence with their characteristic exponents,\nsee e.g. \\cite[Theorems 8.1 and 7.10, and Corollaries 8.3 and 11.6]{MR1739520}, that is\n\\[\n\t\\EE e^{i \\sprod{x}{Y_t}}=e^{-t\\Psi(x)}.\n\\] Non-zero strictly stable processes in $\\RR^d$ form a subclass of L{\\'e}vy processes in $\\RR^d$ indexed\nby $\\alpha\\in(0,2]$. For every non-zero strictly $\\alpha$-stable process $\\mathbf{Y}$ in $\\RR^d$, $\\alpha \\in (0, 2]$,\nthere are unique $A$, $\\lambda$ and $\\gamma$ such that the characteristic exponent $\\Psi$ has the form:\n\\begin{enumerate}[label=\\rm (A.\\roman*), start=1, ref=A.\\roman*]\n\t\\item\n\t\\label{en:4:1}\n\tif $\\alpha=2$, then\n\t\\[\n\t\t\\Psi(x)=\\sprod{x}{Ax}\n\t\\]\n\twhere $A$ is a non-zero $d\\times d$ symmetric non-negative definite real matrix;\n\t\\item\n\t\\label{en:4:2}\n\tif $\\alpha \\in (1, 2)$, then\n\t\\[\n\t\t\\Psi(x)=\\int_{\\mathbb{S}}\n\t\t\\lambda({\\rm d}\\xi)\n\t\t\\int_0^\\infty \\big(1-e^{i\\sprod{x}{r\\xi}} +i \\sprod{x}{r\\xi} \\big) \\frac{{\\rm d}r}{r^{1+\\alpha}}\n\t\\]\n\twhere $\\lambda$ is a non-zero finite measure on the unit sphere $\\mathbb{S}$;\n\t\\item\n\t\\label{en:4:3}\n\tif $\\alpha=1$,\n\t\\[\n\t\t\\Psi(x)=\\int_{\\mathbb{S}}\\lambda({\\rm d}\\xi)\n\t\t\\int_0^\\infty \\big(1-e^{i\\sprod{x}{r\\xi}} + i \\sprod{x}{r\\xi} \\ind{(0,1]}(r)\\big)\n\t\t\\frac{{\\rm d}r}{r^{1+\\alpha}}-i\\sprod{\\gamma}{x}\n\t\\]\n\twhere either $\\gamma\\in\\RR^d$ and $\\lambda$ is a non-zero finite measure on the unit sphere $\\mathbb{S}$ satisfying\n\t\\[\n\t\t\\int_{\\mathbb{S}} \\xi\\, \\lambda({\\rm d}\\xi)=0\n\t\\]\n\tor $\\gamma \\neq 0$ and $\\lambda\\equiv 0$;\n\t\\item\n\t\\label{en:4:4}\n\tif $\\alpha \\in (0, 1)$,\n\t\\[\n\t\t\\Psi(x)=\\int_{\\mathbb{S}}\\lambda({\\rm d}\\xi)\n\t\t\\int_0^\\infty \\big(1-e^{i\\sprod{x}{r\\xi}} \\big) \\frac{{\\rm d}r}{r^{1+\\alpha}}\n\t\\]\n\twhere $\\lambda$ is a non-zero finite measure on the unit sphere $\\mathbb{S}$.\n\\end{enumerate}\nConversely, for every function $\\Psi$ of the form as in \\eqref{en:4:1}--\\eqref{en:4:4} there is a unique non-zero strictly\n$\\alpha$-stable process $\\mathbf{Y}$ in $\\RR^d$ with $\\alpha\\in(0,2]$.\n\nLet us observe that the only non-zero trivial L{\\'e}vy process is a non-zero constant drift, that is\n$\\alpha=1$ with $\\gamma\\neq 0$ and $\\lambda \\equiv 0$. Thus, an alternative representation of characteristic exponents of\nnon-zero strictly stable processes is a consequence of \\cite[Theorem 14.10 and Example 18.8]{MR1739520}. The classification\n\\eqref{en:4:1}--\\eqref{en:4:4} is a summary of the following results: \\cite[Theorem 14.2 and Example 2.10]{MR1739520} for\n$\\alpha=2$ and \\cite[Theorem 14.7, Remarks 14.6 and 14.4]{MR1739520} for $\\alpha \\in (0, 2)$. For precise definitions\nof a strictly stable process see \\cite[Definitions 13.2 and 13.16]{MR1739520}, and a non-zero and nontrivial\nprocess see \\cite[Definition 13.6]{MR1739520}. Let us recall that for a give non-zero strictly $\\alpha$-stable\nprocess, $\\alpha \\in (0, 2]$, the generating triplet is identified as follows:\n\\begin{enumerate}[label=\\rm (A.\\roman*), start=1, ref=A.\\roman*]\n\t\\item\n\tif $\\alpha = 2$, then take $(A, 0, 0)$;\n\t\\item\n\tif $\\alpha \\in (1, 2)$, then take $(0, \\nu, \\gamma)$ where\n\t\\[\n\t\t\\gamma = -\\int_{\\{\\norm{z} > 1\\}} z \\nu({\\rm d} z);\n\t\\]\n\t\\item\n\tif $\\alpha = 1$, then we take $(0, \\nu, \\gamma)$;\n\t\\item\n\tif $\\alpha \\in (0, 1)$, we take $(0, \\nu, \\gamma)$ where\n\t\\[\n\t\t\\gamma = \\int_{\\{ |z| \\leq 1 \\}}z \\nu({\\rm d}z);\n\t\\]\n\\end{enumerate}\nIn view of \\cite[Theorem 14.3]{MR1739520},\n\\begin{equation}\n\t\\label{eq:nu-str_st}\n\t\\nu(B)=\\int_{\\mathbb{S}}\\lambda({\\rm d}\\xi)\n\t\\int_0^\\infty \\ind{B} (r\\xi) \\frac{{\\rm d}r}{r^{1+\\alpha}}, \\qquad B \\in \\mathcal{B}(\\RR^d).\n\\end{equation}\nIn our studies, we shall assume that for each $t > 0$, the distribution of $Y_t$ is absolutely continuous with respect to\nthe Lebesgue measure. In fact, for strictly stable processes, it is equivalent to the absolute continuity of $Y_1$.\n\nThe proof of the following lemma is inspired by \\cite[Proposition 24.20 and Example 37.19]{MR1739520}.\nFor the definition of non-degenerate process see \\cite[Definitions 24.16 and 24.18]{MR1739520}.\n\\begin{lemma}\n\t\\label{lem:density}\n\tLet $\\mathbf{Y}$ be a strictly $\\alpha$-stable process in $\\RR^d$, $\\alpha\\in(0,2]$. The following statements are equivalent:\n\t\\begin{enumerate}[label=\\rm (\\roman*), start=1, ref=\\roman*]\n\t\t\\item\n\t\t\\label{en:1:1}\n\t\tthe distribution of $Y_1$ is absolutely continuous with respect to the Lebesgue measure;\n\t\t\\item\n\t\t\\label{en:1:2}\n\t\t$\\mathbf{Y}$ is non-degenerate;\t\t\\item\n\t\t\\label{en:1:3}\n\t\tif $\\alpha = 2$, then $\\det A \\neq 0$; otherwise for each $x \\in \\RR^d \\setminus \\{0\\}$,\n\t\t\\begin{equation}\n\t\t\t\\label{eq:60}\n\t\t\t\\int_{\\mathbb{S}} |\\sprod{x}{\\xi}| \\: \\lambda({\\rm d}\\xi) \\neq 0;\n\t\t\\end{equation}\n\t\t\\item\n\t\t\\label{en:1:4}\n\t\t$\\Re ( \\Psi(x)) \\approx |x|^{\\alpha}$.\n\t\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n\tEach of the conditions \\eqref{en:1:1}--\\eqref{en:1:4} implies that $\\mathbf{Y}$ is non-zero. Clearly,\n\t\\eqref{en:1:1} $\\implies$ \\eqref{en:1:2} follows from the definition of the non-degeneracy.\n\tNext, let us assume that \\eqref{en:1:2} holds true. If $\\alpha=2$, then by \\cite[Proposition 24.17]{MR1739520} we must\n\thave $\\det(A)\\neq 0$. If $\\alpha\\in(0,2)$, by \\cite[Theorem 14.10 and Example 18.8]{MR1739520}, we have that\n\t\\begin{equation}\n\t\t\\label{eq:59}\n\t\t\\Re (\\Psi(x)) = \\int_{\\mathbb{S}} |\\sprod{x}{\\xi}|^{\\alpha} \\lambda_1({\\rm d}\\xi)\n\t\\end{equation}\n\twhere $\\lambda_1$ is a positive measure proportional to $\\lambda$. Suppose, contrary to our claim, that there is\n\t$x_0 \\in \\RR^d \\setminus\\{0\\}$, such that $\\Re (\\Psi(x_0)) = 0$. Hence, the measure $\\lambda_1$ as well as the\n\tL{\\'e}vy measure $\\nu$, are supported on $x_0^\\perp$. This contradicts the\n\tnon-degeneracy of $\\mathbf{Y}$, see \\cite[Proposition 24.17]{MR1739520}. This proves \\eqref{en:1:2} $\\implies$\n\t\\eqref{en:1:3}.\n\t\n\tLet us observe that for $x \\in \\RR^d \\setminus \\{0\\}$,\n\t\\[\n\t\t\\Re \\Phi(x) = \\norm{x}^\\alpha \\Re \\Psi\\Big(\\tfrac{x}{\\norm{x}} \\Big).\n\t\\]\n\tIndeed, it follows immediately from \\eqref{en:4:1} for $\\alpha = 2$, and from \\eqref{eq:59} for $\\alpha \\in (0, 2)$.\n\tBy the continuity and compactness of the unit sphere, $\\Re \\Phi$ attains its extremes. Therefore, to show that \\eqref{en:1:3}\n\t$\\implies$ \\eqref{en:1:4}, it is enough to prove that the minimum on $\\mathbb{S}$ is positive. For $\\alpha \\in (0, 2)$, it\n\tis an easy consequence of \\eqref{eq:60}. If $\\alpha = 2$, since the matrix $A$ is symmetric and non-negative definite, the\n\tcondition $\\det A \\neq 0$ implies that it is in fact positive definite, proving \\eqref{en:1:4}.\n\t\n\tFinally, \\eqref{en:1:4} $\\implies$ \\eqref{en:1:1} easily follows by the Fourier inversion formula.\n\\end{proof}\n\n\\begin{remark}\n\t\\label{rem:density}\n\tIf $d=1$, the only non-zero strictly $\\alpha$-stable process which is \\emph{degenerate} is a non-zero drift, that is,\n\t$\\alpha=1$, $\\gamma \\neq 0$ and $\\lambda \\equiv 0$. Hence, by Lemma~\\ref{lem:density}, for every non-zero strictly\n\t$\\alpha$-stable process other than drift, $Y_1$ is absolutely continuous with respect to the Lebesgue measure.\n\\end{remark}\nIf any of the conditions \\eqref{en:1:1}--\\eqref{en:1:4} in Lemma \\ref{lem:density} is satisfied, then for each $t > 0$\nthe random variable $Y_t$ has absolutely continuous density $p_0(t; 0, \\cdot)$. Moreover, $p_0(u; 0, \\cdot)$ belongs to\n$\\calC_0^\\infty(\\RR^d)$, and\n\\begin{align}\n\t\\label{eq:p_0}\n\tp_0(t;x,y)= \\bigg(\\frac{1}{2\\pi}\\bigg)^d\n\t\\int_{\\RR^d} e^{-i\\sprod{y-x}{z}} e^{-t \\Psi(z)} {\\: \\rm d}z,\n\\end{align}\nsee \\cite{Hartman, MR3010850, MR4140542} and \\cite[Section 6]{MR3996792} for a broader context.\n\\begin{lemma}\n\t\\label{lem:A3}\n\tLet $\\mathbf{Y}$ be a strictly $\\alpha$-stable process in $\\RR^d$, $\\alpha\\in(0,2]$, with a density $p_0$.\n\t\\begin{enumerate}[label=\\rm (\\roman*), start=1, ref=\\roman*]\n\t\t\\item\n\t\t\\label{en:3:1}\n\t\tFor all $x,y\\in\\RR^d$ and $t,u>0$,\n\t\t\\[\n\t\t\tp_0(tu; x, y)=t^{-d/\\alpha}p_0(u;t^{-1/\\alpha}x,t^{-1/\\alpha}y).\n\t\t\\]\n\t\t\\item\n\t\t\\label{en:3:2}\n\t\tFor every $\\ell \\in \\NN_0$, $\\mathbf{a} \\in \\NN_0^d$ there is a constant $C > 0$ such that for all $y \\in \\RR^d$\n\t\tand $u>0$,\n\t\t\\[\n\t\t\t\\partial_u^\\ell \\,\\partial_y^\\mathbf{a} p(u;0,\\cdot) \\in \\calC_0^\\infty\\big(\\RR^d\\big),\n\t\t\\]\n\t\tand\n\t\t\\[\n\t\t\t\\big| \\partial_u^\\ell \\partial_y^\\mathbf{a} p(u;0,y)\\big|\n\t\t\t\\leq C u^{-\\ell-(d+|\\mathbf{a}|)/\\alpha}.\n\t\t\\]\n\t\t\\item\n\t\t\\label{en:3:3}\n\t\tThere is a constant $C > 0$ such that for all $x,y,z\\in\\RR^d$ and $u>0$,\n\t\t\\begin{align*}\n\t\t\tp_0(u;0,y-x)\n\t\t\t\\leq C u^{-d/\\alpha},\n\t\t\\end{align*}\n\t\tand\n\t\t\\begin{align*}\n\t\t\t\\big|p_0(u;0,y+z)-p_0(u;0,y)\\big|\n\t\t\t\\leq C |z| u^{-(d+1)/\\alpha}.\n\t\t\\end{align*}\n\t\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n\tThe proof of \\eqref{en:3:1} follows from \\eqref{eq:p_0} and the scaling property of $\\Psi$, that is\n\t$t\\Psi(z)=\\Psi(t^{1/\\alpha}z)$. By Lemma~\\ref{lem:density} and \\cite[Proposition~2.17]{MR3156646} we obtain that\n\t$\\Re\\Psi(z) \\approx |z|^{\\alpha}$ and $|\\Psi(z)|\\leq c (1+|z|^2)$, respectively. Thus, we can differentiate under\n\tthe integral sign in \\eqref{eq:p_0}, and the integrands are absolutely integrable. This proves the regularity in\n\t\\eqref{en:3:2}. Now using the scaling of $\\Psi$ we get\n\t\\[\n\t\t\\partial_u^{\\ell} \\partial_y^\\mathbf{a} p_0(u;0,y)=\n\t\tu^{-\\ell-(d+|\\mathbf{a}|)/\\alpha}\\big(\\partial_u^\\ell \\,\\partial_y^\\mathbf{a} \\, p_0\\big)\\big(1;0,u^{-1/\\alpha}y\\big)\n\t\\]\n\twhich together with the above mentioned estimates on $\\Psi$ leads to the upper bound in \\eqref{en:3:2}.\n\tFinally, \\eqref{en:3:3} easily follows from \\eqref{en:3:2}.\n\\end{proof}\n\n\\begin{example}\n\t\\label{ex:names:1}\n\tLet $\\mathbf{Y}$ be an $\\alpha$-stable subordinator with $\\alpha\\in (0,1)$, and let $d = 1$. If\n\t$\\lambda({\\rm d}\\xi)= \\frac{\\alpha}{\\Gamma\\left(1-\\alpha\\right)} \\delta_{1}({\\rm d}\\xi)$, then\n\t\\[\n\t\t\\nu({\\rm d}s)=\\frac{\\alpha}{\\Gamma\\left(1-\\alpha\\right)} \\ind{(0, \\infty]}(s) \\frac{{\\rm d} s}{s^{1+\\alpha}}.\n\t\\]\n\tMoreover, the Laplace transform is $\\EE e^{- u Y_t}= e^{-t u^{\\alpha}}$, $u \\geq 0$. See \\cite[Example 24.12]{MR1739520}.\n\\end{example}\n\n\\begin{example}\n\t\\label{ex:names:2}\n\tLet $\\mathbf{Y}$ be an isotropic $\\alpha$-stable process with $\\alpha\\in (0,2)$. If\n\t$\\lambda({\\rm d}\\xi)=c \\lambda_0({\\rm d}\\xi)$ (and $\\gamma=0$ for $\\alpha=1$) where $\\lambda_0$ is the uniform probability\n\tmeasure on the unit sphere $\\mathbb{S}$, and\n\t\\[\n\t\tc^{-1}=c_0(d,\\alpha)=2^{-\\alpha}\\frac{\\Gamma(d/2)\\Gamma((2-\\alpha)/2)}{\\alpha \\Gamma((\\alpha+d)/2)}.\n\t\\]\n\tThen\n\t\\[\n\t\t\\nu({\\rm d}z)= \\mathscr{A}_{d,\\alpha} |z|^{-d-\\alpha} {\\: \\rm d}z,\n\t\t\\quad\\text{ and }\\quad\n\t\t\\mathscr{A}_{d,\\alpha} =\\frac{2^\\alpha \\Gamma((d+\\alpha)/2)}{\\pi^{d/2} |\\Gamma(-\\alpha/2)|},\n\t\\]\n\tand $\\EE e^{i\\sprod{x}{Y_t}}=e^{-t|x|^\\alpha}$. See \\cite[Example 18.9]{MR1739520}.\n\\end{example}\n\n\\begin{example}\n\t\\label{ex:names:3}\n\tLet $\\mathbf{Y}$ be a cylindrical $\\alpha$-stable process with $\\alpha\\in (0,2)$. If $d \\geq 2$, then\n\t\\[\n\t\t\\lambda ({\\rm d\\xi})=\\sum_{k=1}^d \\Big( \\frac{c}{2} \\big(\\delta_{-1}({\\rm d}\\xi_k)\n\t\t+\\delta_{1}({\\rm d}\\xi_k)\\big) \\prod_{\\substack{j=1 \\\\ j\\neq k}}^d \\delta_0 ({\\rm d}\\xi_j)\\Big)\n\t\\]\n\twhere $c^{-1} = c_0(1, \\alpha)$. If $\\alpha = 1$, we also have $\\gamma = 0$. Then\n\t\\[\n\t\t\\nu({\\rm d}z) = \\sum_{k=1}^d \\Big( \\mathscr{A}_{1,\\alpha} |z_k|^{-1-\\alpha} {\\: \\rm d}z_k\n\t\t\\prod_{\\substack{j=1 \\\\ j \\neq k}}^d \\delta_0 ({\\rm d} z_j) \\Big),\n\t\\]\n\tand $\\EE e^{i\\sprod{x}{Y_t}}=\\exp\\{-t (|x_1|^\\alpha+\\ldots +|x_d|^\\alpha)\\}$.\n\\end{example}\n\n\\begin{example}\n\tLet $\\mathbf{Y}$ be Brownian motion. If $A=I$ (identity matrix), then\n\t\\[\n\t\tp_0(t;x,y)=(4\\pi t)^{-d/2}e^{-\\frac{|y-x|^2}{4t}},\n\t\\]\n\tand $\\EE e^{i\\sprod{x}{Y_t}} = \\exp\\{-t \\norm{x}^2\\}$.\n\\end{example}\n\nThere is also a formula for the transition density of $\\frac{1}{2}$-stable subordinator, see e.g.\n\\cite[Example 2.13]{MR1739520}. Namely,\n\\[\n\tp_0(t;x,y)= (4\\pi)^{-1/2} t (y-x)^{-3/2} e^{- \\frac{t^2}{4(y-x)}} \\ind{\\{x<y\\}}.\n\\]\nAnother example covered in the article with an explicit formula for the transition density\nis the Cauchy process, i.e., a L{\\'e}vy process in $\\RR^d$ with the characteristic function\n\\[\n\t\\EE e^{i\\sprod{x}{Y_t}}=e^{-t(|x|- i \\sprod{\\gamma}{x})}.\n\\]\nFor $\\gamma=0$, the formula reduces to the isotropic $1$-stable process, called the \\emph{symmetric Cauchy process}. For general\n$\\gamma\\in\\RR^d$ it is a strictly $1$-stable process with the density, see e.g. \\cite[Example 2.12]{MR1739520},\n\\[\n\tp_0(t;x,y)= \\frac{\\Gamma\\big(\\tfrac{d+1}{2}\\big)}{\\pi^{\\frac{d+1}{2}}}\n\t\\frac{t}{\\big(|y-x-\\gamma t|^2+t^2\\big)^\\frac{d+1}{2} }.\n\\]\n\n\\section{Further comments}\n\\label{appendix:B}\n\\subsection{Literature overview}\nResetting describes an evolutionary pattern encountered in a variety of physical phenomena (see \\cite{Gupta, MR4093464} for\nsurvey). Apart from the topics already described in Introduction, see Section \\ref{sec:Intro}, it models reset algorithms\n(see e.g. \\cite{Avrach, MR2023017}), and appears as the fluid limit for some queuing models with binomial catastrophe\nrates (see e.g. \\cite{Artalejo}). Such processes can be viewed as particular examples of the so-called shot noise model.\nThey are used in modeling earthquakes or layers of sedimentary material accumulated in depositional environments,\nbut not subject to subsequent erosion (see e.g. \\cite{18}). It can also model snow avalanches or neuron firings (see e.g.\n\\cite{MR1780992, MR2213972, MR2044928,MR1102872}). Applications of resetting procedure have also been discussed\nin the context of backtrack recovery in RNA polymerase (see e.g. \\cite{4, 9}), enzymatic velocity (see e.g. \\cite{16}),\npollination strategies (see e.g. \\cite{17}), enzymatic inhibition (see e.g. \\cite{2}), stochastic thermodynamics\n(see e.g. \\cite{21}), and quantum mechanics (see e.g. \\cite{22}). Realization of resetting has also been confirmed\nin switching holographic optical tweezers (see \\cite{8} for details). Partial resetting also models disasters and\ncatastrophes (see e.g. \\cite{MR4189343}), for example, in the context of population dynamics\n(see e.g. \\cite{Gerber, Assaf, 33, 34}). The risk process with partial resetting found to be an indispensable part of the so-called microinsurance policies\n(see e.g. \\cite{Corina}). In \\cite{Rev} it has been pointed out that a process with resetting can be used to model\ncertain chemical reactions. Additionally, the partial resetting mechanism has also been observed in granular gas where collisions\nconvert the relative speed of two particles quasi-instantaneously to its fractions (see e.g. \\cite{27}), and in a soil\ncontaminant where rainfall events (or other events) leach the contaminant to its fraction\n(see e.g. \\cite{28, 29,30, 18}). Other possible applications of the partial resetting procedure\ninclude vegetation biomass with fires (see e.g. \\cite{31, 32}), modeling of financial crashes (see, e.g.\n\\cite{35, 36, Mendoza, Sor}), or queueing with random clearing events (see e.g. \\cite{38}).\n\nMuch attention has been devoted to the fact that resetting can produce non-equilibrium steady states (see e.g.\n\\cite{Eule, Malakar, Gupta, Evans, Evans1, MR4093464, MR3476293, Pal, White, frenchNESS}). In most of the papers, the density\nof the stationary measure of the process with resetting is described either numerically or in explicit form (e.g. in the case\nof Brownian motion with total resetting). Then there are arguments given to prove that so-called balance conditions are\nviolated. Roughly, the conditions check if the reversed in time process with resetting seen from the point of view of\ninvariant law is the same as original process starting at the invariant law.\nIn \\cite{frenchNESS}, in the context of NESS, the authors discussed the duality of the generator for symmetric exclusion process\nwith reservoirs.\n\n\\subsection{Fourier transforms}\nLet $Y$ be the isotropic strictly stable process. Let $\\mathbf{X}$ be obtained from $\\mathbf{Y}$ by partial resetting\nwith factor $c \\in (0, 1)$. Let $p$ denote a transition density of the process $\\mathbf{X}$. Following \\cite{jaifizycy},\nwe can formally compute the Laplace--Fourier transform of $p$ as well as the Fourier transform of the density\n$\\rho_{\\mathbf{Y}}$ of the stationary measure for $\\mathbf{X}$.\n\nIndeed, applying the Laplace--Fourier transform to both sides of the second equation of Theorem \\ref{thm:H+F-P} with\n$\\mathscr{L}^*f(x)=\\mathscr{L}f(x)=\\Delta^{\\alpha/2}f(x)$ (with usual understanding that for $\\alpha=2$ we have\n$\\Delta^{\\alpha/2}=\\Delta$) we obtain\n\\begin{equation*}\n\t\\label{Fp}\n\tr\\mathcal{F}(p)(\\theta, r)-1= -\\mathcal{F}(p)(\\theta, r)\\big(|\\theta|^\\alpha+1\\big)+ \\mathcal{F}(p)(c\\theta, r),\n\t\\quad\\text{for } r\\geq 0, \\theta \\in \\RR\n\\end{equation*}\nwhere\n\\[\n\t\\mathcal{F}(p)(\\theta, r)=\\int_0^\\infty e^{-rt}\\mathcal{F}(p(t; 0,\\cdot ))(\\theta) {\\: \\rm d} t.\n\\]\nHence, it is expected that\n\\begin{equation*}\n\t\\label{Fpa}\n\t\\mathcal{F}(p)(\\theta, r)=\\sum_{k=0}^\\infty \\prod_{j=0}^k \\frac{1}{1+|\\theta|^\\alpha m^{j}+r}.\n\\end{equation*}\nSimilarly, using the first equation in Theorem \\ref{thm:H+F-P} we obtain\n\\begin{equation*}\n\t\\label{Fpi}\n\t-\\mathcal{F}(\\rho_{\\mathbf{Y}})(\\theta)\\big(|\\theta|^\\alpha+1\\big)+ \\mathcal{F}(\\rho_{\\mathbf{Y}})(c\\theta)=0,\n\\end{equation*}\nand thus\n\\begin{equation*}\n\t\\label{Fpb}\n\t\\mathcal{F}(\\rho_{\\mathbf{Y}})(\\theta)=\\prod_{j=0}^\\infty \\frac{1}{1+|\\theta|^\\alpha m^{j}}.\n\\end{equation*}\nNote that from Theorem~\\ref{thm:H+F-P} it follows that\n$\\rho_{\\mathbf{Y}}$ satisfies the pantograph functional-differential equation, which has reach structure\n(see e.g. \\cite{iserles_1993, WISNIEWOLSKI2024600}).\n\n\\subsection{L\\'evy process with partial resetting and autoregressive process of order $1$}\nLet $\\mathbf{Y}$ be a one-dimensional L\\'evy process with a transition density $p_0$. Let $\\mathbf{X}$ be obtained from\n$\\mathbf{Y}$ by partial resetting with factor $c \\in (0, 1)$. Then the density of the stationary distribution of $\\mathbf{X}$\nexists and it equals the stationary distribution of the process $\\mathbf{X}$ observed at Poisson epochs only.\nIndeed, according to \\cite[Definition 1]{MR1157423}, $\\mathbf{X}$ admits an embedded recursive chain\n$\\mathbf{Z}=(Z_n =X_{T_n}: n \\in \\NN_0)$ being an autoregressive process of order $1$ defined via\n\\[\n\t\\begin{cases}\n\t\tZ_0 = 0, \\\\\n\t\tZ_{n+1} =c Z_n + Y_{T_{n+1}-T_n}, &\\quad n \\in \\NN.\n\t\\end{cases}\n\\]\nBy \\cite[Theorem 1(3), Theorem 8 and Theorem 2]{MR1157423}, it is enough to show that $\\mathbf{Z}$ convergence in total\nvariation towards its stationary law. The latter follows from \\cite[Theorem 2.1 and Remark B]{MR1956829} and\n\\cite[Theorem 2.8]{MR1894253}, since $\\mathbf{Z}$ is an aperiodic, positive Harris recurrent Markov chain that has an\nabsolutely continuous distribution (see also \\cite[Theorem 13.0.1]{MR1287609}).\n\nFurthermore, denoting by $X_\\infty$ the random variable with distribution being the invariant law of $\\mathbf{X}$,\nwe have the following affine equation satisfied\n\\begin{equation}\n\t\\label{affine}\n\tX_\\infty \\eqdistr cX_\\infty+ Y_{T_1}.\n\\end{equation}\nIn particular, if $\\mathbf{Y}$ is a one-dimensional strictly isotropic $\\alpha$-stable process, then by \\cite{Grinc} and\n\\cite[Lemma A.3]{MikoschSamrodnitsky}, as $y\\rightarrow +\\infty$, we get\n\\begin{enumerate}\n\t\\item if $\\alpha \\in (0, 2)$, then\n\t\\begin{align*}\n\t\t\\mathbb{P} (\\pm X_\\infty >y)\n\t\t&\\sim (1-c^{\\alpha})^{-1}\\mathbb{P}( \\pm Y_{T_1}>y) \\\\\n\t\t&\\sim \\frac{1}{2\\alpha \\Gamma(-\\alpha)\\cos \\left(\\frac{(2-\\alpha)\\pi}{2}\\right)}y^{-\\alpha},\n\t\t\\qquad \\text{as } y \\rightarrow + \\infty;\n\t\\end{align*}\n\t\\item if $\\alpha = 2$, then\n\t\\[\n\t\t\\mathbb{P} (X_\\infty >y) \\sim \\frac{1}{(m,m)_\\infty}\\mathbb{P} (Y_{T_1} >y)\n\t\t\\qquad\\text{as } y \\rightarrow +\\infty.\n\t\\]\n\\end{enumerate}\n\n\\subsection{Generalized Ornstein--Uhlenbeck process and exponential functional}\nObserve that the one-dimensional process $\\mathbf{X}$ obtained from the L\\'evy process $\\mathbf{Y}$\nby partial resetting is a solution of the stochastic differential equation\n\\begin{equation}\n\t\\label{sde1}\n\t\\mathrm{d}X_t=(c-1)X_{t-}\\;\\mathrm{d}N_t+{\\rm d} Y_t,\n\\end{equation}\nhence it is a special case of the so-called Generalized Ornstein--Uhlenbeck process. Therefore,\n$X_\\infty$ being the solution of \\eqref{affine} has the same law as the law of the following exponential functional\n\\[\n\tD=\n\t\\int_0^\\infty e^{-\\log (1+c)N_{s-}}\\mathrm{d} Y_s,\n\\]\nsee e.g. \\cite{Behme}.\n\n\\section*{Acknowledgement}\nThe second author acknowledges that the research is partially supported by the Polish National Science Centre\nGrant No.~2021/41/B/HS4/00599.\n\\section*{Data availability}\nWe do not analyse or generate any datasets, because our work proceeds within a theoretical and mathematical approach.\n\\section*{Conflict of interest}\nOn behalf of all authors, the corresponding author states that there is no conflict of interest. \n\\begin{thebibliography}{10}\n\n\\bibitem{Corina}\nJ.~Akahori, C.~Imamura~Y. Constantinescu, and H.~Pham, \\emph{An application of\n risk theory to mortgage lending}, Scand. Actuar. 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Zolotarev, \\emph{One-dimensional stable distributions}, Translations of\n Mathematical Monographs, vol.~65, American Mathematical Society, Providence,\n RI, 1986.\n\n\\end{thebibliography}\n\n\\end{document}\n\n\n\n\n"}}},{"rowIdx":199,"cells":{"id":{"kind":"string","value":"2412.15688v1"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.15688v1"},"title":{"kind":"string","value":"On the number of connected edge cover sets in a graph"},"full_text":{"kind":"string","value":"\\documentclass[11pt]{article}\n\\usepackage{amssymb,amsfonts,amsmath,latexsym,epsf,tikz,url}\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{observation}[theorem]{Observation}\n\\newtheorem{conjecture}[theorem]{Conjecture}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{rem}[theorem]{Remark}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{definition}[theorem]{Definition}\n\n\\newcommand{\\proof}{\\noindent{\\bf Proof.\\ }}\n\\newcommand{\\qed}{\\hfill $\\square$\\medskip}\n\n\\textwidth 14.5cm\n\\textheight 21.0cm\n\\oddsidemargin 0.4cm\n\\evensidemargin 0.4cm\n\\voffset -1cm\n\n\\begin{document}\n\\title{On the number of connected edge cover sets in a graph }\n\n\\author{Mahsa Zare$^1$ \\and Saeid Alikhani$^{1,}$\\footnote{Corresponding author} \\and Mohammad Reza Oboudi$^2$}\n\n\\date{\\today}\n\n\\maketitle\n\n\\begin{center}\n\n $^1$Department of Mathematical Sciences, Yazd University, 89195-741, Yazd, Iran\\\\\n{\\tt zare.zakieh@yahoo.com~~~alikhani@yazd.ac.ir}\\\\\n$^2$Department of Mathematics, College of Science, Shiraz University, Shiraz, Iran\n{\\tt mr\\_oboudi@shirazu.ac.ir}\n\\end{center}\n\n\\begin{abstract}\nLet $ G=(V,E) $ be a simple graph of order $ n $ and size $ m $. A connected edge cover set of a graph is a subset $S$ of edges such that every vertex of the graph is incident to at least one edge of $S$ and the subgraph induced by $S$ is connected. \nWe initiate the study of the number of the connected edge cover sets of a graph $G$ with cardinality $i$, $ e_{c}(G,i) $ and consider the generating function for $ e_{c}(G,i) $\nwhich is called the connected edge cover polynomial of $ G $. After obtaining some results for this polynomial, we investigate this polynomial for some certain graphs. \n\\end{abstract}\n\n\\noindent{\\bf Keywords:} Edge cover number, connected edge cover number, cubic graphs. \n\n\\medskip\n\\noindent{\\bf AMS Subj.\\ Class.}: 05C30, 05C69. \n\n\\section{Introduction}\n Let $G=(V,E)$ be a simple graph. The {\\it order} and the size of $G$ is the number of vertices and the number of edges of $G$, respectively.\n For every graph $G$ with no isolated vertex, an edge covering of $G$ is a set of edges of $G$ such that every vertex is incident with at least one edge of the set. In other words, an edge covering of a graph is a set\n of edges which together meet all vertices of the graph. A minimum edge covering is an edge covering of the smallest possible size. The edge covering number of $G$ is the size of a minimum edge covering of\n $G$ and is denoted by $\\rho(G)$. We let $\\rho(G) = 0$, if $G$ has some isolated vertices.\n For a detailed treatment of these parameters, the reader is referred to~\\cite{saeid1,JAS,bond,GRo}.\n Let $\\mathcal{E}(G,i)$ be the family of all edge coverings of a graph $G$ with cardinality $i$ and\n let $e(G,i)=|{\\mathcal{E}}(G,i)|$.\n The { edge cover polynomial} $E(G,x)$ of $G$ is defined as\n \\[\n E(G, x)=\\sum_{ i=\\rho(G)}^{m} e(G, i) x^{i},\n \\]\n where $\\rho(G)$ is the edge covering number of $G$. Also, for a graph $G$ with some isolated vertices\n we define $E(G, x) = 0$. Let $E(G, x) = 1$, when\n both order and size of $G$ are zero (see \\cite{saeid1}).\n \n In \\cite{saeid1} authors have characterized all graphs whose edge cover polynomials have exactly one or two distinct roots and moreover they proved that these roots\n are contained in the set $\\{-3,-2,-1, 0\\}$. In \\cite{JAS}, authors constructed some infinite families of graphs whose\n edge cover polynomials have only roots $-1$ and $0$. Also, they studied the edge coverings and edge\n cover polynomials of cubic graphs of order $10$. As a consequence , they have shown that the all cubic graphs of order $10$ (especially the\n Petersen graph) are\n determined uniquely by their edge cover polynomials.\n\nMotivated by the edge cover number, we consider the following definition. \n\n\\begin{definition}\n\tA {\\it connected edge cover set} of graph $G$ is a subset $S$ of edges such that every vertex of $G$ is incident to at least one edge of $S$ and the subgraph induced by $S$ is connected. The connected edge cover number of $G$, $ \\rho_{c}(G)$, is the minimum cardinality of the connected edge cover.\n\\end{definition}\n\nAlso, we state the following definition for the connected edge cover polynomial. \n\n\\medskip\n\n\n\n\\begin{definition}\t\n\t\tThe {\\it connected edge cover polynomial} of $ G $ is the polynomial \n\t\\[\n\t E_{c}(G,x)=\\sum_{i=1}^{m} e_{c}(G,i)x^{i},\n\t \\]\n\t where $ e_{c}(G,i) $ is the number of connected edge cover set of size $ i $.\n\\end{definition}\n\n\nFor two graphs $G$ and $H$, the corona $G\\circ H$ is the graph arising from the\ndisjoint union of $G$ with $| V(G) |$ copies of $H$, by adding edges between\nthe $i$th vertex of $G$ and all vertices of $i$th copy of $H$. The corona $G\\circ K_1$, in particular, is the graph constructed from a copy of $G$, where for each vertex $v\\in V(G)$, a new vertex $u$ and a pendant edge $\\{v, u\\}$ are added.\nIt is easy to see that the corona operation of two graphs does not have the commutative property.\n\n\\medskip\n\nUsually the generalized friendship graph is denoted by $ F_{n,m} $ which is a collection of $ n $ cycles (all of order $ m$), meeting at a common vertex.\n\n\\medskip\n\nTwo graphs $ G $ and $ H $ are said to be connected edge covering equivalent, or simply {\\it ${\\mathcal{E}_{c}}$-equivalent}, written $ G\\sim_{c}H $, if $ E_{c}(G,x)=E_{c}(H,x) $. It is evident that the relation $\\sim_{c}$ of being\n${\\mathcal{E}_{c}}$-equivalence is an equivalence relation on the family ${\\cal G}$ of graphs, and thus ${\\cal G}$ is partitioned into equivalence classes, called the {\\it ${\\mathcal{E}_{c}}$-equivalence classes}. Given $G\\in {\\cal G}$, let\n\\[\n[G]=\\{H\\in {\\cal G}:H\\sim_{c} G\\}.\n\\]\nWe call $[G]$ the equivalence class determined by $G$.\n\nA graph $ G $ is said to be connected edge covering unique or simply {\\it $ E_{c} $-unique}, if $ [G]={G} $.\n\n\\medskip\n\nIn this paper, we obtain the connected edge cover polynomial for certain graphs. \n\n\\section{Connected edge cover polynomial}\nHere, we state some new results on the connected edge cover number and the connected edge cover polynomial. The following theorem is easy to obtain: \n\n\\begin{theorem}\n\nFor every natural number $ n\\geq 3 $,\n\\begin{enumerate}\n\t\\item [(i)]\n\t$ E_{c}(K_{n},x)=E(K_{n},x)-\\sum_{ i=\\lceil n/2\\rceil}^{n-2} e(K_{n}, i) x^{i} $.\n\t\n\t\n\t\\item[(ii)]\n\tFor every natural number $ n\\geq 3 $, \n\t$ \\rho_{c}(C_{n})=n-1 $ and $ E_{c}(C_{n},x)=\\sum_{ i=n-1}^{n} {n \\choose i} x^{i} $.\n\t\\item[(iii)] \nFor every natural number $ n\\geq 5 $,\n$ E_{c}(P_{n},x)= x^{n-1} $.\n\\end{enumerate}\n\\end{theorem}\n\n\n\n\\medskip\n\n\\begin{theorem}\nFor every natural numbers $n$ and $ m\\geq 3$,\n$ E_{c}(F_{n,m},x)=\\sum_{i=0}^{n} {n \\choose i} m^{i} x^{mn-i} $.\n\\end{theorem}\n\\begin{proof}\nWe know that $\\Delta(F_{n,m})=mn$ and $\\delta(F_{m,n})=n(m-1)$.\nTo construct a connected edge cover set of $F_{m,n}$ with cardinal $ mn-i$, it is enough to choose $ m-1 $ edges from $ m $ edges of $i$ cycles $C_m$. So $e_c(F_{m,n},mn-i)={n \\choose i} m^{i}$ and so we have the result. \\qed\n\\end{proof}\n\n\n\n\n\n\\begin{theorem}\nIf $ G $ is a graph with order $ n $ and $ E_{c}(G ,x)=E_{c}(K_{n} ,x) $, then $ G=K_{n} $.\n\\end{theorem}\n\n\n\\begin{proof}\nSince the degree of $ E_{c}(K_{n} ,x) $ is $m=\\frac{n(n-1)}{2}$ and $ E_{c}(G ,x)=E_{c}(K_{n},x) $, so $ G $ is a graph of size $ m $. On the other hand, the only connected graph of the order $ n $ and size $ m=\\frac{n(n-1)}{2}$ is graph $ K_{n} $. Therefore $ G=K_{n} $.\\qed\n\\end{proof}\n\n\n\nHere, we obtain an recursive formula for the connected edge cover polynomial of graphs.\nLet $u\\in V(G)$. By\n$N_u$ we mean the set of all edges of $G$ incident with $u$.\n\n\\begin{theorem}\\label{main}\n\nLet $ G $ be a graph, $ u, v\\in V(G) $ and $ uv\\in E(G) $. Then\n$$ E_{c}(G, x)=(x+1)E_{c}(G\\setminus uv, x)+xE_{c}(G\\setminus v, x)+xE_{c}(G\\setminus u, x) .$$\n\\end{theorem}\n\n\\begin{proof}\nIf $G$ has an isolated vertex, then $G$ is a disconnected graph, so there is nothing to prove. Suppose that $ \\delta(G)\\geq1 $ and $ S $ is a connected edge covering set of $ G $ of size $ i $.\n\\begin{itemize}\n\t\\item \nIf $ uv\\notin S $, then we have two cases:\n\n\\begin{enumerate}\n\n\\item[(1)] $ deg(v)=1 $ or $ deg(u)=1 $. So $ S $ is a disconnected graph.\n\n\\item[(2)] $ deg(v)>1 $ and $ deg(u)>1 $. So $ S $ is a connected edge covering set of $ G\\setminus uv $ with size $ i $.\n\\end{enumerate}\n\n\n\\item If $ uv\\in S $, then we have the following cases:\n\\begin{enumerate}\n\n\n\\item[(i)] \n$ |S\\cap N_{u}|=|S\\cap N_{v}|=1 $. So in this case $ S $ is disconnected graph.\n\n\\item[(ii)] \n$ |S\\cap N_{u}|>1 $ and $|S\\cap N_{v}|=1 $. Therefore $ S\\setminus uv $ is a connected edge covering set of $ G\\setminus v $ with size $ i-1 $.\n\n\\item[(iii)]\n $|S\\cap N_{u}|= 1 $ and $|S\\cap N_{v}|>1 $. Therefore $ S\\setminus uv $ is a connected edge covering set of $ G\\setminus u $ with size $ i-1 $.\n\n\\item[(iv)] \n$|S\\cap N_{u}|>1 $ and $|S\\cap N_{v}|>1 $. Therefore $ S\\setminus uv $ is a connected edge covering set of $ G\\setminus uv $ with size $ i-1 $.\n\n\\end{enumerate}\n\\end{itemize} \nSo we have\n$$ e_{c}(G, i)= e_{c}(G\\setminus uv, i)+ e_{c}(G\\setminus v, i-1)+ e_{c}(G\\setminus u, i-1)+ e_{c}(G\\setminus uv, i-1), $$\n\nand so we have the result. \\qed \n\\end{proof} \n\n\\medskip\n\nBy Theorem \\ref{main}, we have the following corollary:\n\n\\begin{corollary}\n\t\\begin{enumerate} \n\\item[(i)] \nFor every natural number $ n\\geq 3 $,\n$ E_{c}(P_{n}, x)= xE_{c}(P_{n-1}, x) $.\n\n\\item[(ii)] \nFor every natural number $ n\\geq 4 $,\n$ E_{c}(C_{n}, x)= xE_{c}(C_{n-1}, x)+x^{n-1} $.\n\n\\end{enumerate}\n\\end{corollary}\n\n\n\nHere, we consider the connected edge cover number and the connected edge cover polynomial for corona of some graphs. \n\n\n\\begin{theorem} \n\t\\begin{enumerate}\n\t\t\\item [(i)]\n\t\tFor any connected graph $ G $ of order $ n $, $ \\rho_{c}(G\\circ K_{1})=2n-1$.\n\t\t\n\t\t\\item[(ii)] \n\t\tFor any natural number $ n\\geq3 $, and for every $ i $, $ 2n-1\\leq i\\leq n+\\frac{n(n-1)}{2}$, $$ e_{c}(K_{n}\\circ K_{1}, i)={\\frac{n(n-1)}{2} \\choose i-n}-n{n-1 \\choose i-n} .$$\n\t\\end{enumerate}\n\n\\end{theorem}\n\\begin{proof}\n\t\t\\begin{enumerate}\n\t\t\t\\item [(i)]\nIf $ S $ is a connected edge covering of $ G\\circ K_{1} $, then $S$ contains at least $ n-1 $ edges of the graph $ G $ and $ n $ edges which connect the vertices of $G$ and the copies of graph $ K_{1} $. So we have $|S|\\geq 2n-1$ and so we have the result. \n\n\\item[(ii)] \nAny edge cover set of $ K_{n}\\circ K_{1} $ of size $ i $ should contain $n$ edges of the outer $C_n$. Now we should choose $i-n$ edges from any $n$ induced subgraph $K_{n-1}$. Therefore, we have the result. \\qed\n\\end{enumerate} \n \\end{proof}\n\n\n\n\n\n\\medskip\n\n\n\\begin{theorem} \nLet $ G $ be a connected graph of order $ n $ and size $ m $. If $ E_{c}(G,x)=\\sum_{i=1}^{m} e_{c}(G,i)x^{i} $, then the following hold:\n\\begin{enumerate}\n\\item[(i)] $ E_{c}(G, x) $ is a monic polynomial of degree $ m $.\n\n\\item[(ii)] $ n\\leq \\rho_{c}(G)+1 $.\n\n\\item[(iii)] For $ i\\geq m-\\delta+1 $, $ e_{c}(G, i)={m \\choose i} $. Moreover, if $ i_{0}=min \\lbrace i \\vert e_{c}(G, i)={m \\choose i}\\rbrace $, then $ \\delta=m-i_{0}+1 $.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n\\begin{enumerate}\n\n\n\\item[(i)] Since $ E(G) $ is the unique connected edge covering of $ G $ of size $ m $, so the result follows. \n\n\\item[(ii)] Since any $ n-1 $ edges in graph $G$ is a connected edge covering of $ G $, so we have the result.\n\n\\item[(iii)] Let $ i\\geq m-\\delta+1 $. So every subset $S\\subseteq E(G)$ of size $i$ is a connected edge covering of $G$.\n Now, suppose that $i \\leq m-\\delta$. Consider a vertex $v$ of degree $\\delta$. Let $A\\subseteq \\overline{N_v}$, such\nthat $|A|=i$. Clearly, $A$ is not a connected edge covering of $G$. So $e_c(G,i)<{m\\choose i}$. \\qed\n\n\\end{enumerate}\n\\end{proof}\n\n\n\\medskip\n\n\n\\begin{corollary} \nLet $ G $ and $ H $ be two connected graphs of size $ m_{1} $ and $ m_{2} $. If $ E_{c}(H, x)=E_{c}(G, x) $, then $ \\rho_{c}(G)=\\rho_{c}(H) $, $ m_{1}=m_{2} $ and $ \\delta(G)=\\delta(H) $.\n\n\\end{corollary}\n\n\n\n\\medskip\n\n\\section{Cubic graphs of order $6$, $8$ and the Petersen graph}\nIn this section, we compute the number of connected edge cover set of size $ \\rho_{c} $ for cubic graphs of order $6$, $8$ and the Petersen graph. Domination polynomials of cubic graphs of order $10$ has studied in \\cite{turk} and the Coalition of cubic graphs of order at most $10$ studied in \\cite{CCO}. \nThe cubic graphs of order $6$ has shown in Figure \\ref{1}. \n\\medskip\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[scale=0.8]{C6}\n\t\\caption{Cubic graphs of order 6} \\label{1} \n\\end{figure}\n\nThe following results give $e_c(G_1, \\rho_{c}(G_1))$ and $e_c(G_2, \\rho_{c}(G_2))$ for the cubic graphs of order $6$. \n\n\\begin{theorem} \\label{cub6}\n\t\t $ e_{c}(G_{1},5)= e_{c}(G_{2}, 5)=81$.\n\\end{theorem}\n\\begin{proof}\n\t\tConsider the graphs $G_1$ and $G_2$ in Figure \\ref{1}. To construct a connected edge covering set $S$ of size $5$: \n\n\n\\noindent $\\bullet$\n Choose $5$ edges from the cycle \n$ \\{ \\{ 1,2 \\},\\{ 2,3 \\},\\{ 3,4 \\},\\{ 4,5 \\},\\{ 5,6 \\},\\{ 6,1\\} \\}$ in Figure \\ref{1}. So we have $6$ distinct sets. \n\n\\noindent $\\bullet$\n Choose $4$ edges from the cycle $ \\{ \\{ 1,2 \\},\\{ 2,3 \\},\\{ 3,4 \\},\\{ 4,5 \\},\\{ 5,6 \\},\\{ 6,1\\} \\} $ and one another edge that one of its end-vertex is a vertex which is not on the $4$ chosen edges. So we have $ {6 \\choose 4}{1 \\choose 1}=15 $ distinct connected edge covering set.\n\n\\noindent $\\bullet$\n Choose $3$ edges from the cycle $ \\{ \\{ 1,2 \\},\\{ 2,3 \\},\\{ 3,4 \\},\\{ 4,5 \\},\\{ 5,6 \\},\\{ 6,1\\} \\} $ and $2$ edges from $ \\{ \\{ 1,4 \\}, \\{ 2,6 \\}, \\{ 3,5 \\} \\} $, except for the case that $3$ edges of the cycle $ \\{ \\{ 1,2 \\}, \\{ 2,3 \\},\\\\ \\{ 3,4 \\},\\{ 4,5 \\},\\{ 5,6 \\},\\{ 6,1 \\} \\} $ are connected. So in case, we have \n$ {6 \\choose 3}{3 \\choose 2}-{6 \\choose 1}\\times2=48 $ distinct connected edge covering set.\n\n\\noindent \n$\\bullet$\n Choose $3$ edges from $ \\{ \\{ 1,4 \\}, \\{ 2,6 \\}, \\{ 3,5 \\}\\} $ and $2$ edges from $ \\{ \\{ 1,2 \\},\\{ 2,3 \\},\\{ 3, \\\\ 4 \\},\\{ 4,5 \\},\\{ 5,6 \\},\\{ 6,1\\} \\} $, except for three states $ \\{ \\{\\{1,2\\},\\{6,1\\}\\}, \\{\\{2,3\\},\\{5,6\\}\\}, \\{\\{3,4\\},\\\\\\{4 ,5\\}\\} \\} $. So in case we have\n $ {3 \\choose 3}\\times [{6 \\choose 2}-3]=12 $ distinct connected edge covering set.\n\nTherefore, by the addition principle, $e_{c}(G_{1},5)=81$. \\qed\n\n\\end{proof}\n\nSimilar to the proof of Theorem \\ref{cub6}, we can compute another coefficients of cubic graphs of order $6$ and we have the following result: \n\n\\begin{theorem}\nIf $G_1$ and $G_2$ are two cubic graphs of order $6$ (Figure \\ref{1}), then \n\n$$ E_{c}(G_{1}, x)=E_{c}(G_{2}, x)=x^{9}+{9 \\choose 8}x^{8}+{9 \\choose 7}x^{7}+{9 \\choose 6}x^{6}+81x^{9}.$$\n\\end{theorem}\n\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[scale=0.8]{C8}\n\t\\caption{Cubic graphs of order 8} \\label{2} \n\\end{figure} \n\nHere, we obtain the number of connected edge covering sets of size $\\rho_c$ of cubic graphs of order $8$ which have shown in Figure \\ref{2}. \n\n\\begin{theorem}\\label{cube8}\n\t\\begin{enumerate}\n\t\t\\item[(i)] $ e_{c}(G_{1},7)=324$.\n\t\t\\item[(ii)]\n\t\t$ e_{c}(G_{2}, 7)=338 $.\n\t\t\\item[(iii)] \n\t\t$ e_{c}(G_{3}, 7)= e_{c}(G_{4}, 7)=332 $.\n\t\t\\item[(iv)]\n\t\t $ e_{c}(G_{5}, 7)=344 $.\n\t\t\\end{enumerate}\n \\end{theorem}\n\\begin{proof}\nThe proof of all parts are easy and similar. For instance, we state the proof of Part \n(i). To construct a connected edge covering of size $7$ of $G_1$, we have some cases:\n\\begin{enumerate}\n\n\\item[(1)] Choose seven edges from $ \\{ \\{1,2\\}, \\{2,3\\}, \\{3,4\\}, \\{4,5\\}, \\{5,6\\}, \\{6,7\\}, \\{7,8\\}, \\{8,1\\} \\} $. So we have ${8 \\choose 7}=8$ distinct connected edge covering sets.\n\n\\item[(2)] Choose six edges from $ \\{ \\{1,2\\}, \\{2,3\\}, \\{3,4\\}, \\{4,5\\}, \\{5,6\\}, \\{6,7\\}, \\{7,8\\}, \\{8,1\\} \\} $ and one another edge that one of its end-vertices is a vertex which is not on the $6$ chosen edges. So we have $ {8 \\choose 6}{1 \\choose 1}=28 $ distinct connected edge covering sets.\n\n\n\\item[(3)] Choose five edges from $ \\{ \\{1,2\\}, \\{2,3\\}, \\{3,4\\}, \\{4,5\\}, \\{5,6\\}, \\{6,7\\}, \\{7,8\\}, \\{8,1\\} \\} $. We have the following subcases:\n\n\\textbf{Case 1}. Choose five edges of induced subgraph $ P_{6} $ from the cycle $ \\{ \\{1,2\\}, \\{2,3\\},\\\\ \\{3,4\\}, \\{4,5\\}, \\{5,6\\}, \\{6,7\\}, \\{7,8\\}, \\{8,1\\} \\} $, and choose $2$ edges from $2$ edges that are connected to vertices which are not on the $5$ chosen edges. So, we have $ 8\\times {2 \\choose 2}=8 $ distinct connected edge covering sets.\n\n\\textbf{Case 2}. Choose five edges from outer $C_8$ such that a vertex of $C_8$, say $v$ is not incident to chosen edges. Then we can choose two edges of $ \\{ \\{1,2\\}, \\{2,3\\},\\\\ \\{3,4\\}, \\{4,5\\}, \\{5,6\\}, \\{6,7\\}, \\{7,8\\}, \\{8,1\\} \\} $ such that the distance between an end-vertex and the vertex $v$ is two. So, we choose one of the edges with end-vertex $v$ and one of the edges whose induced subgraph have leaf. Therefore in this case there are\n$ {8 \\choose 1}{1 \\choose 1}{4 \\choose 3}{1 \\choose 1}=32 $ connected edge cover sets.\n \n\\textbf{Case 3}. If the induced subgraph of $5$ chosen edges does not have an isolated vertex, then we subtract two previous cases from the total states and we know that two pairs vertices from $6$ vertices with degree one have repetitive edges connecting to each other, so we have $4$ vertices and we need to choose $2$ edges connected to them. Thus in this case there are\n$ [{8 \\choose 5}-(8+32)]\\times {4 \\choose 2}=96 $ connected edge cover sets.\n\n\\item[(4)] Choose four edges from $ \\{ \\{1,2\\}, \\{2,3\\}, \\{3,4\\}, \\{4,5\\}, \\{5,6\\}, \\{6,7\\}, \\{7,8\\}, \\{8,1\\} \\} $. We have the following subcases:\n\n\\textbf{Case 1}. Consider an isolated vertex (in the induced subgraph), say $v$. Choose one of the edges with end-vertex $v$, one edge from edges that are adjacent with an edge with end-vertex $v$ and one of the edges from two edges that is adjacent to an edge $K_2$ in induced subgraph. So in this case, there are \n$ {8 \\choose 1}{1 \\choose 1}{1 \\choose 1}{2 \\choose 1}=16 $ connected edge cover sets.\n\n\\textbf{Case 2}. Choose two isolated vertices (in the induced subgraph), two vertices from 2 vertices that are connected to isolated vertices and one of edge from 2 other edges. Thus in this case there are\n$ {8 \\choose 2}{2 \\choose 2}{2 \\choose 1}=56 $ connected edge cover sets.\n\n\\textbf{Case 3}. Choose there isolated vertices (in the induced subgraph) and there vertices from 3 vertices that are connected to isolated vertices. Therefore in this case there are \n$ {8 \\choose 3}{3 \\choose 3}=56 $ connected edge cover sets.\n\n\\item[(5)] Choose four edges from $\\{ \\{1,7\\}, \\{2,4\\}, \\{3,5\\}, \\{6,8\\} \\} $. We have the following subcases:\n\n\\textbf{Case 1}. Choose two edges from $ \\{ \\{1,2\\}, \\{5,6\\} \\} $ and one edge from other 6 edges. So in this case there are \n$ {4 \\choose 4}{2 \\choose 2}{6 \\choose 1}=6 $ connected edge cover sets.\n\n\\textbf{Case 2}. Choose one edge from $ \\{ \\{1,2\\}, \\{5,6\\} \\} $, one edge from $ \\{ \\{2,3\\}, \\{3,4\\}, \\{4,5\\} \\} $ and one edge from $ \\{ \\{6,7\\}, \\{7,8\\}, \\{8,1\\} \\} $. So in this case there are\n$ {4 \\choose 4}{2 \\choose 1}{3 \\choose 1}{3 \\choose 1}=18 $ connected edge cover sets.\n\nAccording the addition principle, $e_{c}(G_{1}, 7)=324 $. \\qed\n\n\\end{enumerate}\n\\end{proof} \n\nSimilar to the proof of Theorem \\ref{cube8}, we can obtain another coefficients of the connected edge cover polynomial of cubic graphs of order $8$ and so we have the following result: \n\n\\begin{theorem}\nIf $ G_{1},...,G_{5}$ are the cubic graphs of oder $8$ (Figure \\ref{2}), then \n\n\\item[(i)] $ E_{c}(G_{1}, x)=x^{12}+{12 \\choose 11}x^{11}+[{12 \\choose 7}-1]x^{10}+[{12 \\choose 9}-6]x^{9}+[{12 \\choose 8}-6]x^{8}+324x^{7} $,\n\n\\item[(ii)] $ E_{c}(G_{2}, x)=x^{12}+{12 \\choose 11}x^{11}+{12 \\choose 7}x^{10}+{12 \\choose 9}x^{9}+[{12 \\choose 8}-2]x^{8}+338x^{7} $,\n\n\\item[(iii)] $ E_{c}(G_{3}, x)=E_{c}(G_{4}, x)=x^{12}+{12 \\choose 11}x^{11}+{12 \\choose 7}x^{10}+{12 \\choose 9}x^{9}+[{12 \\choose 8}-4]x^{8}+332x^{7} $,\n\n\\item[(iv)] $ E_{c}(G_{5}, x)=x^{12}+{12 \\choose 11}x^{11}+{12 \\choose 7}x^{10}+{12 \\choose 9}x^{9}+{12 \\choose 8}x^{8}+344x^{7} $.\n\n\\end{theorem}\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[scale=1]{P2}\n\t\\caption{The Petersen graph} \\label{Pet} \n\\end{figure}\n\n\n\n\\medskip\n\n\nThere are exactly $21$ cubic graphs of order $10$ (\\cite{JAS}), of them all, the Petersen graph $P$ is the most well-known. Here, we compute the number of connected edge cover sets of $P$ of size $9$. \n\n\\begin{lemma} \n If $P$ is the Petersen graph, then $ e_{c}(P, 9)=235$.\n\n\\end{lemma}\n\n\n\n\\begin{proof}\nTo construct a connected edge covering of $P$ of size $9$ (Figure \\ref{Pet}), we have some cases:\n\\begin{enumerate}\n\n\n\\item[(i)] Choose four edges from the cycle $ \\{ \\{1,2\\},\\{2,3\\},\\{3,4\\},\\{4,5\\},\\{5,1\\} \\} $, one edge from $ \\{ \\{1,6\\}, \\{2,7\\}, \\{3,8\\}, \\{4,9\\}, \\{5,10\\} \\} $ and four edges from $ \\{ \\{6,8\\}, \\{6,9\\}, \\{7,9\\}, \\\\ \\{7,10\\}, \\{8,10\\} \\} $. So in case, there are $ {5 \\choose 4}{5 \\choose 1}{5 \\choose 4}=125 $ connected edge cover sets.\n\n\n\\item[(ii)] Choose four edges from the cycle $ \\{ \\{1,2\\},\\{2,3\\},\\{3,4\\},\\{4,5\\},\\{5,1\\} \\} $, four edges from $ \\{ \\{1,6\\}, \\{2,7\\}, \\{3,8\\}, \\{4,9\\}, \\{5,10\\} \\} $ and there are four ways to connect an isolated vertex to other vertices. Therefore in case, there are $ {5 \\choose 4}{5 \\choose 4}\\times 4=100 $ connected edge cover sets.\n\n\\item[(iii)] Choose five edges from $ \\{ \\{1,6\\}, \\{2,7\\}, \\{3,8\\}, \\{4,9\\}, \\{5,10\\} \\} $ and four edges from $ \\{ \\{6,8\\}, \\{6,9\\}, \\{7,9\\}, \\{7,10\\}, \\{8,10\\} \\} $. So in this\ncase there are $ {5 \\choose 5}{5 \\choose 4}=5 $ connected edge cover sets.\n\n\n\n\\item[(iv)] Choose four edges from the cycle $ \\{ \\{1,2\\},\\{2,3\\},\\{3,4\\},\\{4,5\\},\\{5,1\\} \\} $ and five edges from $ \\{ \\{1,6\\}, \\{2,7\\}, \\{3,8\\}, \\{4,9\\}, \\{5,10\\} \\} $. So in this\ncase there are $ {5 \\choose 4}{5 \\choose 5}=5 $ connected edge cover sets.\n\n\\end{enumerate}\n\nTherefore, by the addition principle, we have the result. \\qed \n\n\\end{proof}\n\n\n\n\\section{Conclusion} \nIn this paper we considered the connected edge cover sets of a graph and initiated the study of the number of the connected edge cover sets with cardinality $i$. We introduced the connected edge cover polynomial of a graph and found some of its properties. We computed this polynomial for cubic graphs of order $6,8$ and also we computed the number of connected edge cover sets of the Petersen graphs of size $9$. There are many open problems for future works. We state some of them:\n\n\\begin{enumerate}\n\t\\item [(i)]\n\tStudy the roots of the connected edge cover polynomial of a graph. \n\t\n\t\\item[(ii)] \n\tWhat are the formula for the connected edge cover polynomial of some graph products, such as, Cartesian, corona, join and lexicographic. \n\t\n\t\\item[(iii)] \n\tExamine the effects on $\\rho(G), \\rho_c(G)$ and $E_c(G,x)$ when $G$ is modified by operations \ton vertex and edge of $G$. \n\t \n\\end{enumerate} \n\n\n\n\n\\begin{thebibliography}{9}\n\n\\bibitem{saeid1} S. Akbari, M.R. Oboudi, {\\it On the edge cover polynomial of a graph}, { Europ. J. Combin.} {34} (2013) 297--321.\n\n\n\\bibitem{JAS} S. Alikhani, S. Jahari, {\\it On the edge cover polynomials of certain graphs}, {J. Alg. Sys.} 2 (2) (2014) 97--108.\n\n\\bibitem{turk} S. Alikhani, Y.H, Peng, {\\it Domination polynomials of cubic graphs of order 10}, Turk. J. Math. 35 (2011) 355-366.\n\n\n\\bibitem{CCO} S. Alikhani, H.R. Golmohammadi, E.V. Konstantinova, {\\it Coalition of cubic graphs of order at most $10$}, Commun. Comb. Optim. 9 (3) (2024) 437-450.\n \n\\bibitem{bond} J.A. Bondy, U.S.R. Murty, {\\it Graph Theory}, Graduate Texts in Mathematics, 244.\nSpringer, New York, 2008.\n\n\n\n\\bibitem{GRo} M. Grohe, D. Marx, {\\it Constraint solving via fractional edge covers}, in: Proceedings of the Seventeenth Annual ACM SIAM Symposium on Discrete Algorithm, SODA 06, New York, NY, USA, 2006, pp. 289--298.\n\n\n\n\n\n\n\\end{thebibliography}\n\n\\end{document}\n\n"}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":1,"numItemsPerPage":100,"numTotalItems":591,"offset":100,"length":100}},"jwt":"eyJhbGciOiJFZERTQSJ9.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc3NzQwNjU4OCwic3ViIjoiL2RhdGFzZXRzL2V0aHotc3B5bGFiL21hdGhfbGF0ZXgiLCJleHAiOjE3Nzc0MTAxODgsImlzcyI6Imh0dHBzOi8vaHVnZ2luZ2ZhY2UuY28ifQ.hXX4lrk68n0byZVzGtdvnjCKXP2qLMiLNh3PuvHcJJ3F2DeIL5ccUVmrM2FHx7neiOtz8W3_e8Xj4P4yQFvkDw","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":591,"numBytesParquet":27690937}]},"discussionsStats":{"closed":0,"open":0,"total":0},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
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2412.11227v2 | http://arxiv.org/abs/2412.11227v2 | The Brascamp-Lieb inequality in Convex Geometry and in the Theory of Algorithms | \documentclass{amsart}
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2412.11225v1 | http://arxiv.org/abs/2412.11225v1 | Cohomology of the diffeomorphism group of the connected sum of two generic lens spaces | \pdfoutput=1
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2412.11283v1 | http://arxiv.org/abs/2412.11283v1 | Cyclic polytopes through the lens of iterated integrals | \documentclass{ourlematema}
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2412.11358v1 | http://arxiv.org/abs/2412.11358v1 | Enumerating Diagonalizable Matrices over $\mathbb{Z}_{p^k}$ | \documentclass{article}
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2412.11415v4 | http://arxiv.org/abs/2412.11415v4 | Delone sets associated with badly approximable triangles | \documentclass[reqno]{amsart}
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2412.11406v1 | http://arxiv.org/abs/2412.11406v1 | Normal surface singularities of small degrees | \documentclass[12pt,reqno]{amsart}
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2412.11508v1 | http://arxiv.org/abs/2412.11508v1 | Legendre theorems for certain overpartitions and overpartition pairs | \documentclass[reqno]{amsart}
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2412.11485v2 | http://arxiv.org/abs/2412.11485v2 | Inexact Proximal Point Algorithms for Zeroth-Order Global Optimization | \documentclass[final,onefignum,onetabnum]{siamart220329}
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2412.11473v1 | http://arxiv.org/abs/2412.11473v1 | Optimal interpolation in Hardy and Bergman spaces: a reproducing kernel Banach space approach | \documentclass[reqno]{amsart}
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2412.11547v1 | http://arxiv.org/abs/2412.11547v1 | Weak convergence of complex Monge-Ampère operators on compact Hermitian manifolds |
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2412.11528v1 | http://arxiv.org/abs/2412.11528v1 | Water Cells in Compositions of 1s and 2s | \documentclass[11pt]{amsart}
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2412.11646v1 | http://arxiv.org/abs/2412.11646v1 | BA-BFL: Barycentric Aggregation for Bayesian Federated Learning | \documentclass{article}
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2412.11607v1 | http://arxiv.org/abs/2412.11607v1 | Nonlocal double phase Neumann and Robin problem with variable $s(\cdot,\cdot)-$order | \documentclass[reqno,a4paper,11pt]{amsart}
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2412.11697v1 | http://arxiv.org/abs/2412.11697v1 | On the second coefficient in the semi-classical expansion of Toeplitz Operators on CR manifolds | \documentclass[12pt]{amsart}
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2412.11833v1 | http://arxiv.org/abs/2412.11833v1 | A monotone block coordinate descent method for solving absolute value equations | \documentclass[12pt,a4paper,twoside]{article}
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2412.11841v1 | http://arxiv.org/abs/2412.11841v1 | A Serrin-type over-determined problem for Hessian equations in the exterior domain |
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2412.11935v1 | http://arxiv.org/abs/2412.11935v1 | Riesz Bases in Krein Spaces |
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2412.11933v2 | http://arxiv.org/abs/2412.11933v2 | Generalised Fermat equation: a survey of solved cases | \documentclass[11pt]{article}
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2412.16193v1 | http://arxiv.org/abs/2412.16193v1 | Arithmetic properties of $k$-tuple $\ell$-regular partitions | \documentclass[12pt, reqno]{amsart}
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2412.11999v2 | http://arxiv.org/abs/2412.11999v2 | Pattern-avoiding shallow permutations | \documentclass{article}
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2412.12070v1 | http://arxiv.org/abs/2412.12070v1 | Simultaneous and multiplicative Diophantine approximation on missing-digit fractals | \documentclass[12pt,reqno]{amsart}
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2412.12372v1 | http://arxiv.org/abs/2412.12372v1 | On Mahler's conjecture for even s-concave functions in dimensions 1 and 2 | \documentclass[11pt,reqno]{amsart}
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2412.12608v1 | http://arxiv.org/abs/2412.12608v1 | SOR-like iteration and FPI are consistent when they are equipped with certain optimal iterative parameters | \documentclass[]{interact}
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2412.12607v1 | http://arxiv.org/abs/2412.12607v1 | Linear Convergence of Resolvent Splitting with Minimal Lifting and its Application to a Primal-Dual Algorithm | \documentclass[10pt]{article}
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2412.12633v2 | http://arxiv.org/abs/2412.12633v2 | Arborescences of Random Covering Graphs | \documentclass[submission]{FPSAC2025}
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2412.18623v2 | http://arxiv.org/abs/2412.18623v2 | Total restrained coalitions in graphs |
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2412.12854v1 | http://arxiv.org/abs/2412.12854v1 | Non-Uniqueness Phase in Hyperbolic Marked Random Connection Models using the Spherical Transform | \documentclass[12pt,a4paper,dvipsnames]{article}
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2412.12921v2 | http://arxiv.org/abs/2412.12921v2 | Equivariant and Invariant Parametrized Topological Complexity | \documentclass[12pt,reqno,a4paper]{amsart}
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2412.12958v1 | http://arxiv.org/abs/2412.12958v1 | The exact subgraph hierarchy and its local variant for the stable set problem for Paley graphs | \pdfoutput=1
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2412.13011v1 | http://arxiv.org/abs/2412.13011v1 | All non-Gaussian states are advantageous for channel discrimination: Robustness of non-convex continuous variable quantum resources | \documentclass[aps,pra,groupedaddress,bibnotes,twocolumn,reprint]{revtex4-2}
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2412.13075v1 | http://arxiv.org/abs/2412.13075v1 | Monogenic Cyclic Cubic Trinomials | \documentclass[12]{amsart}
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2412.13068v1 | http://arxiv.org/abs/2412.13068v1 | Regularity lost: the fundamental limitations and constraint qualifications in the problems of elastoplasticity | \documentclass{article}
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2412.13056v1 | http://arxiv.org/abs/2412.13056v1 | Profiniteness of higher rank volume | \documentclass[12pt, a4paper]{amsart}
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2412.13118v1 | http://arxiv.org/abs/2412.13118v1 | Entanglement principle for the fractional Laplacian with applications to inverse problems | \documentclass[12pt,final]{amsart}
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2412.13424v1 | http://arxiv.org/abs/2412.13424v1 | Remarks on retracts of polynomial rings in three variables in any characteristic |
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2412.13427v2 | http://arxiv.org/abs/2412.13427v2 | Spectrality of Moran-type measures with staggered contraction ratios | \documentclass[12pt, reqno]{amsart}
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2412.13557v1 | http://arxiv.org/abs/2412.13557v1 | The $L_p$ Gauss dual Minkowski problem | \documentclass[12pt]{amsart}
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2412.13583v1 | http://arxiv.org/abs/2412.13583v1 | Spectrum and Lifshitz tails for the Anderson model on the Sierpinski gasket graph | \documentclass[11pt,reqno]{amsart}
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2412.13634v1 | http://arxiv.org/abs/2412.13634v1 | Kinetically constrained models | \documentclass[graybox,envcountchap,sectrefs,envcountsame]{svmono}
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2412.13613v1 | http://arxiv.org/abs/2412.13613v1 | Some New Non-binary Quantum Codes from One-generator Quasi-cyclic Codes | \documentclass[11pt, a4paper]{article}
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2412.13675v1 | http://arxiv.org/abs/2412.13675v1 | On the algebraic structure of the Schröder monoid | \UseRawInputEncoding
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2412.13754v1 | http://arxiv.org/abs/2412.13754v1 | Optimal Exact Recovery in Semi-Supervised Learning: A Study of Spectral Methods and Graph Convolutional Networks | \pdfoutput=1
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2412.13751v1 | http://arxiv.org/abs/2412.13751v1 | Notions of entropy for unitary representations | \documentclass[11pt,a4paper]{book}
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2412.13748v1 | http://arxiv.org/abs/2412.13748v1 | Physical Layer Security for Continuous-Aperture Array (CAPA) Systems | \documentclass[journal]{IEEEtran}
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2412.13759v1 | http://arxiv.org/abs/2412.13759v1 | Iterated relation systems on Riemannian manifolds | \documentclass[12pt,reqno]{amsart}
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2412.13906v1 | http://arxiv.org/abs/2412.13906v1 | Whitney Numbers of Rank-Metric Lattices and Code Enumeration | \documentclass[11pt,a4paper,reqno]{article}
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2412.13898v1 | http://arxiv.org/abs/2412.13898v1 | On consistent estimation of dimension values | \documentclass[11pt,a4paper]{article}
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2412.13929v2 | http://arxiv.org/abs/2412.13929v2 | A novel necessary and sufficient condition for the stability of $2\times 2$ first-order linear hyperbolic systems | \documentclass[10pt]{article}
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2412.13926v1 | http://arxiv.org/abs/2412.13926v1 | Finite groups with coprime non-linear codegrees | \documentclass{amsart}
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2412.13975v1 | http://arxiv.org/abs/2412.13975v1 | The number of descendants in a preferential attachment graph |
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2412.14059v1 | http://arxiv.org/abs/2412.14059v1 | Bessel functions and Weyl's law for balls and spherical shells | \documentclass[11pt]{amsart}
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2412.14125v2 | http://arxiv.org/abs/2412.14125v2 | $η$-Ricci solitons and $η$-Einstein metrics on weak $β$-Kenmotsu $f$-manifolds | \documentclass[11pt,a4paper,twoside]{article}
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\evensidemargin=-4 true mm
\setlength{\textheight}{247 true mm} \setlength{\textwidth}{166 true mm}
\newtheorem{theorem}{Theorem}\newtheorem{corollary}{Corollary}\newtheorem{definitio... |
2412.14277v1 | http://arxiv.org/abs/2412.14277v1 | Quadratically enriched binomial coefficients over a finite field |
\documentclass{amsproc}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage{tikz}
\usetikzlibrary{decorations.markings}
\usepackage{tikz-cd}
\usepackage{slashed}
\usepackage{braket}
\usepackage{extarrows}
\usepackage[margin=1.3in]{geometry}
\usepackage{textcomp}
\usepackage{mathtools}
\use... |
2412.14266v1 | http://arxiv.org/abs/2412.14266v1 | Stability analysis of geodesics in dynamical Chern-Simons black holes: a geometrical perspective |
\documentclass[pdflatex,sn-mathphys-num]{sn-jnl}
\usepackage{graphicx}\usepackage{multirow}\usepackage{amsmath,amssymb,amsfonts}\usepackage{amsthm}\usepackage{mathrsfs}\usepackage[title]{appendix}\usepackage{xcolor}\usepackage{textcomp}\usepackage{manyfoot}\usepackage{booktabs}\usepackage{algorithm}\usepackage{algor... |
2412.14345v1 | http://arxiv.org/abs/2412.14345v1 | Coxeter-type quotients of surface braid groups | \documentclass[12pt,british,reqno]{amsart}
\usepackage{babel}
\usepackage[font=small,format=plain,labelfont=bf,up,textfont=it,up]{caption}
\usepackage[utf8]{inputenc}
\newcommand{\fecho}[1]{\ensuremath{\langle\langle #1 \rangle\rangle}}
\usepackage{cancel}
\usepackage{listings}
\usepackage{faktor}
\usepack... |
2412.14410v1 | http://arxiv.org/abs/2412.14410v1 | The proper geometric dimension of the mapping class group of an orientable surface with punctures | \documentclass[a4paper,12pt,leqno]{amsart}
\input{Preamble}
\usepackage{verbatim}
\begin{document}
\title[]{The proper geometric dimension of the mapping class group of an orientable surface with punctures}
\author[N. Colin]{Nestor Colin}
\author[R. Jiménez Rolland]{Rita Jiménez Rolland}
\author[P. L. León Álvar... |
2412.14389v1 | http://arxiv.org/abs/2412.14389v1 | Rigidity of the hyperbolic marked energy spectrum and entropy for $k$-surfaces | \documentclass[a4paper,11pt,reqno]{amsart}
\usepackage[utf8]{inputenc}
\RequirePackage[l2tabu, orthodox]{nag}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{amsthm,amssymb,amsmath}
\usepackage{latexsym}
\usepackage{graphicx}
\usepackage{subfigure}
\usepackage{tikz-cd}
\usetikzlibrary{arrows}
\usetikzlibrary... |
2412.14423v2 | http://arxiv.org/abs/2412.14423v2 | Cross-Validation with Antithetic Gaussian Randomization | \documentclass[11pt]{article}
\newcommand{\blind}{1}
\usepackage[letterpaper, left=1.2truein, right=1.2truein, top = 1.2truein, bottom = 1.2truein]{geometry}
\usepackage[blocks, affil-it]{authblk}
\usepackage[toc,page]{appendix}
\RequirePackage{amsthm,amsmath,amsfonts,amssymb, enumitem}
\RequirePackage[authoryea... |
2412.14709v1 | http://arxiv.org/abs/2412.14709v1 | Minimal rank of primitively $n$-universal integral quadratic forms over local rings | \documentclass[11pt]{amsart}
\usepackage{amssymb, amsmath, latexsym, bm, mathrsfs}
\usepackage{multirow, xcolor, float}
\usepackage[shortlabels]{enumitem}
\usepackage{tasks}
\usepackage[numbers,sort&compress]{natbib}
\usepackage{soul}
\sodef\lightspacing{}{.09em}{.2em plus .2em}{.5em plus .1em minus .1em}
\renewc... |
2412.14792v1 | http://arxiv.org/abs/2412.14792v1 | The asymptotic estimation of prime ideals in imaginary quadratic fields and Chebyshev's bias |
\documentclass[11pt]{article}
\usepackage[utf8]{inputenc}
\usepackage{graphicx} \usepackage{amsmath,amsfonts,amssymb,amsthm}
\usepackage[mathscr]{eucal}
\usepackage{amscd}
\usepackage{tikz}
\usepackage{tkz-graph}
\usepackage{multicol}
\usepackage{float}
\usepackage{authblk}
\pagestyle{headings}
\newtheorem{theorem}{... |
2412.14813v2 | http://arxiv.org/abs/2412.14813v2 | Solutions of stationary McKean-Vlasov equation on a high-dimensional sphere and other Riemannian manifolds | \documentclass[12pt]{article}
\input{header-2}
\usepackage{graphicx} \usepackage[pdftex]{pict2e}
\newcommand\ANDRE[2][]{{\color{orange}{\textbf{#1}}#2}}
\let\AS\ANDRE
\newcommand\ASpar[2][]{\marginpar{\color{orange}{\textbf{#1}}#2}}
\newcommand\ANNA[2][]{{\color{blue}{\textbf{#1}}#2}}
\renewcommand{\#}{\sharp}
\new... |
2412.15096v2 | http://arxiv.org/abs/2412.15096v2 | Castelnuovo-Mumford regularity of finite schemes | \documentclass[12pt,reqno]{amsart}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{amscd}
\usepackage{xypic}
\usepackage{hyperref}
\textwidth=16.00cm \textheight=22cm \topmargin=0.00cm
\oddsidemargin=0.00cm \evensidemargin=0.00cm \headheight=14.4pt
\headsep=1cm \numberwithin{equation}{secti... |
2412.15366v1 | http://arxiv.org/abs/2412.15366v1 | Capacity and PAPR Analysis for MIMO Faster-than-Nyquist Signaling with High Acceleration | \documentclass[journal]{IEEEtran}
\IEEEoverridecommandlockouts
\usepackage{makecell}
\usepackage{cite}
\usepackage{amsmath,amssymb,amsfonts}
\DeclareMathOperator*{\argmax}{arg\,max}
\usepackage{algorithmic}
\usepackage{graphicx}
\usepackage{textcomp}
\usepackage{xcolor}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{... |
2412.15561v1 | http://arxiv.org/abs/2412.15561v1 | Spirals, Tic-Tac-Toe Partition, and Deep Diagonal Maps | \documentclass[12pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage{amsfonts,latexsym,amsthm,amssymb,amsmath,amscd,esint,dsfont}
\usepackage{graphicx}
\graphicspath{ {./figure/} }
\usepackage[margin=10pt,font=small,labelfont=bf]{caption}
\newcommand{\incfig}[2]{
\def\svgwidth{#2}
\input{figure/#1.pdf_... |
2412.15626v1 | http://arxiv.org/abs/2412.15626v1 | Stationary states for stable processes with partial resetting | \documentclass[11pt]{amsart}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsxtra}
\usepackage{dsfont}
\usepackage{color}
\usepackage[compress, sort]{cite}
\usepackage{enumitem}
\usepackage{graphicx}
\usepackage[type1]{newtxtext}
\usepackage{newtxmath}
\usepackage[english,polish]{... |
2412.15688v1 | http://arxiv.org/abs/2412.15688v1 | On the number of connected edge cover sets in a graph | \documentclass[11pt]{article}
\usepackage{amssymb,amsfonts,amsmath,latexsym,epsf,tikz,url}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{observation}[theorem]{Observation}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheo... |
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