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156d4a8c10b6017232e9d74745d32f47c8d289d9 | subsection | 44 | 53 | Convergence analysis for RapDual | By the optimality condition of the \hat{\ell }-th subproblem (REF ), there exists some \lambda ^* such that&\nabla \psi ^{\hat{\ell }}({\textbf {x}}_*^{\hat{\ell }})+\mathbf {A}^{\top }\lambda ^*\in -N_{X}({\textbf {x}}_*^{\hat{\ell }}),\\
& \nabla \psi _m^{\hat{\ell }}(x_{m^*}^{\hat{\ell }})+\lambda ^*=0, \\
&\mathbf ... | {
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Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
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cce0761166ff3d209f61eed0f60af41fc815885e | subsection | 45 | 53 | Convergence analysis for RapDual | In view of (REF ) and Lemma
REF , we have\mathbb {E}\left[d(\nabla f^{\hat{\ell }}({\textbf {x}}_*^{\hat{\ell }})+\mathbf {A}^{\top }\lambda ^*, -N_{X}({\textbf {x}}_*^{\hat{\ell }}))\right]^2&\le \mathbb {E}\Vert 2\mu ({\textbf {x}}_*^{\hat{\ell }}-\bar{\textbf {x}}^{\hat{\ell }-1})\Vert ^2\\
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"Guanghui Lan",
"Yu Yang"
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b28c92e9c9caa6edcdfaca1f8d1713d37563e05f | subsection | 46 | 53 | Numerical experiments | In this section, we report some preliminary numerical results for both RapGrad and RapDual and demonstrate their potential advantages
in Subsection REF and REF , respectively. | {
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dd8c9c6b9a1ba379fc3bfb1bb59eefb81f14f93f | subsection | 47 | 53 | Nonconvex multi-block optimization | We consider the following compressed sensing problem to test the performance of RapDual:\begin{aligned}\min _{x_i\in X_i} &~ \textstyle {\sum }_{i=1}^m \mathcal {P}_{\lambda ,\gamma ,\epsilon }(x_i)\\
\text{s.t. } &~\textstyle {\sum }_{i=1}^mA_ix_i= b,
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3f55eb148a125cc37e5346a6b37023d287c31868 | subsection | 48 | 53 | Nonconvex multi-block optimization | As we can see, the total number of primal block updates needed to yield a good solution, in terms of both objective value and feasibility, can be much smaller when inner loops are terminated early.
[Figure: Batch and randomized versions of Algorithm on compressed sensing problem with smoothed SCAD objective. Left Figur... | {
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10fa7835a7093d8051ddfdac6d869c388e1f3d15 | subsection | 49 | 53 | Concluding remarks | In this paper, we propose a new randomized accelerated proximal-gradient (RapGrad) method for solving nonconvex finite-sum problems (REF ) and a new randomized primal-dual gradient (RapGrad) method for nonconvex multi-block problems (REF ), respectively. We demonstrate that for problem (REF ) with large condition numbe... | {
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b0c7fa6b253e0cfc2e30c2095fe17e1b6ad471f8 | subsection | 50 | 53 | Proof of Lemma | By convexity of \psi and optimality of x^*, we have~Q_t:&=~\varphi (x^t)+\psi (x^*)+\langle \nabla \psi (x^*), x^t-x^*\rangle -\left[\varphi (x^*)+\textstyle {\tfrac{1}{m}\sum _{i=1}^m}(\psi _i(\hat{\underline{x}}^t_i)+\langle \nabla \psi _i(\hat{\underline{x}}^t_i), x^*-\hat{\underline{x}}^t_i\rangle )\right]\\
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2e3cdec71f214dac86ca73f04c2a53c176e49dd2 | subsection | 51 | 53 | Proof of Lemma | By using the above inequality, (REF ) and (REF ), we obtain0\le & ~\mathbb {E}_s\left[\gamma _1\eta _1 V_\varphi (x^*,x^{0})-\gamma _s(1+\eta _s)V_\varphi (x^*,x^s)\right]+\gamma _s\mathbb {E}_s \left[\textstyle {\tfrac{1}{m}\sum _{i=1}^m}\langle \nabla \psi _{i}(\underline{x}_{i}^s)-\nabla \psi _i(x^*), x^{s-1}-x^s\ra... | {
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28346ffbff546941ee9509c8fd7e8c696c0f7454 | subsection | 52 | 53 | Proof of Lemma | This completes the proof.For any t\ge 1, since ({\textbf {x}}^*, y^*) is a saddle point of (REF ), we have\psi (\hat{{\textbf {x}}}^t) - \psi ({\textbf {x}}^*) + \langle \mathbf {A}\hat{{\textbf {x}}}^t-\mathbf {b},y^*\rangle - \langle \mathbf {A}{\textbf {x}}^*-\mathbf {b}, y^t\rangle + h(y^*)- h(y^t)\ge 0.For nonnega... | {
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87880f5a8eafab223c51bd7e9bc7a5f6ca7819cb | abstract | 0 | 22 | Abstract | Let $\Omega \subset \mathbb{R}^N$ be a bounded domain and $\delta(x)$ be the
distance of a point $x\in \Omega$ to the boundary. We study the positive
solutions of the problem $\Delta u +\frac{\mu}{\delta(x)^2}u=u^p$ in $\Omega$,
where $p>0, \,p\ne 1$ and $\mu \in \mathbb{R},\,\mu\ne 0$ is smaller then the
Hardy constan... | {
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} | 1803.08397 | Positive solutions of semilinear elliptic problems with a Hardy
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"Catherine Bandle",
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c408b18dd11e8c921b9230668c1075ec782471ec | subsection | 1 | 22 | Introduction | In this paper we study positive solutions of problems of the formL_\mu u = \Delta u + \frac{\mu }{\delta (x)^2}u=u^p\quad \mbox{\;{in}\;} \Omega ,where \mu \in \mathbb {R} \setminus \lbrace 0\rbrace , \delta (x) is the distance of a point x\in \Omega to the boundary, p\ne 1 is a positive constant and \Omega \subset \ma... | {
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7dd41bd9b57c53f807665faaeb586d3b9c93b507 | subsection | 2 | 22 | Introduction | More precisely we haveTheorem A. Assume 0<p<1, \mu <1/4, \,\mu \ne 0 and \Omega =B_R:=\lbrace x\in \mathbb {R}^N:|x|<R\rbrace .(i) Problem (REF ) has a unique radial solution u(r) for any u(0)>0.(ii) For any \rho \in (0,R) there exists a unique radial solution of problem (REF ) such thatu(r)=
{\left\lbrace \begin{array... | {
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9e8e65cdeac1da288598274786b5879c997eaf5a | subsection | 3 | 22 | Introduction | For any c \in C^{2+\gamma }(\partial \Omega ), c \ge 0,
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1dc4fcc8ef20344b5545880b82d1ee6d898ce417 | subsection | 4 | 22 | Preliminaries | We recall some lemmas which will be used in the proofs of our theorems.Lemma 2.1 (Maximum principle.) Let \mu < C_H(\Omega ) and \omega \subseteq \Omega . If \Delta u+V_\mu u\ge 0 in \omega and u\in W^{1,2}_0(\omega )
then u\le 0 in \omega .The proof simply follows
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bec79ba856f84416e59e283b633d78f7bc9f29b8 | subsection | 5 | 22 | Preliminaries | If there exist a sub and a supersolution
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fdf387e83a7bf64ab40cbbaa9d267fc882bf5df7 | subsection | 6 | 22 | Ball | Theorem A (i),\,(ii),\,(iii) have been proved in . The existence result (iv) is a consequence of Theorem C. The proof of Theorem B is based on ode techniques partly developed in . | {
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d4127ec05c609f8a30daeb7eea554c934a6cf06c | subsection | 7 | 22 | Proof of Theorem B | The radial solutions of (REF ) in B_R satisfy the ordinary differential equation ( ^{\prime }:= \frac{d}{d r})u^{\prime \prime } +\frac{(N-1)}{r} u^{\prime }
+\frac{\mu }{(R-r)^2}u = u^p \quad \mbox{\;{in}\;} (0,R), \,\,\, u^{\prime }(0)=0\,.This equation has for given u(0)>0 a unique local solution. It can be continue... | {
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fe94bc447d1eb4c889654226fe957cf1d2d98040 | subsection | 8 | 22 | Proof of Theorem B | Then the function\overline{u}(r)= {\left\lbrace \begin{array}{ll}
U_{r_1}(r) &\mbox{\;{in}\;} [0, \tilde{r}],\\
w_+ (r) &\mbox{\;{in}\;} [\tilde{r},R)
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c0a0c576fcbc7f08d35cb4e12156b44b1caaf626 | subsection | 9 | 22 | Proof of Theorem B | It is important to point out that all the local maxima of the solutions of (REF ) are below w_0(r) and the local minima are above w_0(r).The radial symmetry implies that U_r^*(0) = 0 and consequentlyw_r^*(0)= -\frac{2}{p-1}U^*(0) R^{\frac{3-p}{p-1}} < 0\,.Thus w^* decreases in a neighborhood of r=0. If it has a local m... | {
"cite_spans": []
} | 1803.08397 | Positive solutions of semilinear elliptic problems with a Hardy
potential | [
"Catherine Bandle",
"Maria Assunta Pozio"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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3d948ca65276c1ab284ea00946cf9e9dc61b44b2 | subsection | 10 | 22 | Proof of Theorem B | Moreover these results together with (REF ) show that there exists a sequence r_n \nearrow R as n \rightarrow \infty such that\int _{r_0}^{r_n}\frac{\sigma }{(R-s)^2}f(w^*,s)\:ds = \int _{r_0}^{r_n}\frac{s^{N-1}}{(R-s)}f(w^*,s)\:dshas a finite limit for n \rightarrow \infty . Hence f(w^*(R),R)= 0.3. If p > 5, we divide... | {
"cite_spans": []
} | 1803.08397 | Positive solutions of semilinear elliptic problems with a Hardy
potential | [
"Catherine Bandle",
"Maria Assunta Pozio"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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23b18a7b616ae9319ab9fb0853e78df26ba26115 | subsection | 11 | 22 | Proof of Theorem B | Then m \le 1 in B_R and m = 1 on \partial B_R and, since u solves (REF ) in B_R, the differential equation for m(r) isU^*\Delta m + 2 <\nabla m,\nabla U^*>+ m \Delta U^* + \frac{\mu }{\delta ^2}m U^*=m^p (U^*)^p.HenceU^*\Delta m + 2 <\nabla m,\nabla U^*>=m^p (U^*)^p - m (U^*)^p \le 0.The maximum principle implies that ... | {
"cite_spans": []
} | 1803.08397 | Positive solutions of semilinear elliptic problems with a Hardy
potential | [
"Catherine Bandle",
"Maria Assunta Pozio"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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e61ceafbf873f806551e779a7a605999ebd50f65 | subsection | 12 | 22 | Proof of Theorem B | Hence for some \delta _0 >0 we have0<c_0 (R-r)^{\beta _-} \le u(r) \le u_\epsilon (r)\le k (R-r)^{\beta _\epsilon }\,, \,\, \forall r \in (R-\delta _0, R)\,.We replace r by \delta = R-r and consider the function v(\delta ) = u(R-\delta ) \delta ^{-\beta _-} . A straightforward computation leads to(\sigma v^{\prime })^{... | {
"cite_spans": []
} | 1803.08397 | Positive solutions of semilinear elliptic problems with a Hardy
potential | [
"Catherine Bandle",
"Maria Assunta Pozio"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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1e008ba5da490db45a8253358bcdc72177d26e6a | subsection | 13 | 22 | Proof of Theorem B | It is therefore possible to choose \epsilon so small that p(\beta _\epsilon -\beta )+ \beta (p-1) +1>-1, and \beta _\epsilon -\beta >-1. Hence the integrals above converge as \delta \rightarrow 0
and v is uniformly bounded.Next we want to show that v(\delta ) has a limit as \delta tends to 0.If \beta = \beta _- >0, (RE... | {
"cite_spans": []
} | 1803.08397 | Positive solutions of semilinear elliptic problems with a Hardy
potential | [
"Catherine Bandle",
"Maria Assunta Pozio"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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b6dff0ae07ce8d6ad3ee34e57aaa4bf1dfc9aa69 | subsection | 14 | 22 | Proof of Theorem C | The existence of a solution u of (REF ) satisfying (REF ) will be proved by means of a supersolution
\overline{u} and a subsolution \underline{u} of (REF ), which both satisfy (REF ) and which are such that \underline{u} \le \overline{u}.We start with an important observation. For \delta \le \delta _0 \le \rho _0/2, \r... | {
"cite_spans": []
} | 1803.08397 | Positive solutions of semilinear elliptic problems with a Hardy
potential | [
"Catherine Bandle",
"Maria Assunta Pozio"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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c64e9c648396b2ac588b1d316741cdeef1c68f8b | subsection | 15 | 22 | Proof of Theorem C | Then for x \in \Omega _{\delta _0}{\mathcal {A}(\overline{w})} = -A \alpha (1-\alpha ) \delta ^{\alpha -2} +A \alpha \delta ^{\alpha -1} \Delta \delta +\frac{2 \beta }{\delta }<\nabla \delta , \nabla h>\\
+2A \beta \alpha \delta ^{\alpha -2} + \frac{\beta \Delta \delta }{\delta }\overline{w}.Since |\Delta \delta | \le ... | {
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"raw": "C. Miranda, Partial differential equations of elliptic type, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 2, Springer (1970).",
... | 1803.08397 | Positive solutions of semilinear elliptic problems with a Hardy
potential | [
"Catherine Bandle",
"Maria Assunta Pozio"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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d6c4d9596f39c7c9fabcd24eaabe0087e6e1867f | subsection | 16 | 22 | Proof of Theorem C | \hspace{35.0pt} \end{array}\right.Since \mu <0 the function \eta (r) is increasing in a neighborhood of \delta _0/2 and it has no local maximum. Thus \eta (r) is a positive increasing solution in (\delta _0/2, \delta _0).We claim that\lim _{r \rightarrow \delta _0} \frac{\eta (r)}{(\delta _0-r)^{\beta _-}} = C_\eta >0 ... | {
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potential | [
"Catherine Bandle",
"Maria Assunta Pozio"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0c4ba6e894b59807541240be34fdcb160c8385ba | subsection | 17 | 22 | Proof of Theorem C | Following the proof of Lemma 2.4 in , we define w by \eta = \delta ^{\beta _-} v = \delta ^{\beta _+}w. It satisfies the same equation (REF ) as v with \beta _- replaced by \beta _+,\begin{split}
(\sigma _+ w^{\prime })^{\prime }= \beta _+ \sigma _+ \frac{N-1}{(\delta _0-\delta )\delta }w (\ge 0), \\
\mbox{\;{where}\;}... | {
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potential | [
"Catherine Bandle",
"Maria Assunta Pozio"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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95f06e554b5a666987b227804250a3b29829e0aa | subsection | 18 | 22 | Proof of Theorem C | Then the function\tilde{u}(x): = \left\lbrace \begin{array}{c}
M\eta (\delta _0 -\delta (x))\,,\,\,x \in \Omega _{\delta _0/2}\,,
\\
M \,,\,\,x \in \Omega \setminus \Omega _{\delta _0/2}\,.
\end{array}\right.is in C^1(\Omega ) and is a (weak) supersolution of (REF ) satisfying\liminf _{\delta (x, \partial \Omega )\righ... | {
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"raw": "M. Marcus, P.-T. Nguyen, Moderate solutions of semilinear elliptic equations with Hardy potential, Annales de l'Institut Henri Poincare (... | 1803.08397 | Positive solutions of semilinear elliptic problems with a Hardy
potential | [
"Catherine Bandle",
"Maria Assunta Pozio"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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59a1d640cbbc472ebd0e929006588409515ab35e | subsection | 19 | 22 | Proof of Theorem C | Obviouslyz (x) \ge \int _{\partial \Omega \cap B_{\delta (x)/2}(x^*(x))}K^\Omega _\mu (x,y)\,dS_y.The boundary regularity together with the fact that x \in \Omega _{\delta _0}, (\delta _0<\rho _0/2), imply that |\partial \Omega \cap B_{\delta (x^*(x))/2}| \ge \epsilon _0 \delta (x)^{N-1} for some \epsilon _0> 0.
Then b... | {
"cite_spans": []
} | 1803.08397 | Positive solutions of semilinear elliptic problems with a Hardy
potential | [
"Catherine Bandle",
"Maria Assunta Pozio"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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458ce046cafe40e0dcc7be3f6394e72fb09c298c | subsection | 20 | 22 | Proof of Theorem C | Set \underline{w} = (h(x) - a \delta ^\alpha )_+ where \alpha is defined in (REF ), h solves (REF ) and a>0 will be fixed below so that the support of \underline{w} is contained in \Omega _{\delta _0}. This is the case if a \ge a_0 >0 for a suitable a_0. If {\mathcal {A}(\underline{w})} \ge \delta ^{\beta (p-1) }\under... | {
"cite_spans": []
} | 1803.08397 | Positive solutions of semilinear elliptic problems with a Hardy
potential | [
"Catherine Bandle",
"Maria Assunta Pozio"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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9e625c2ebaaf23a5d56cf1b49167cd4b6ca21f22 | subsection | 21 | 22 | Proof of Theorem C | Clearly \underline{h} is a sub–harmonic function, and it satisfies \lim _{x\rightarrow \partial \Omega } \frac{\underline{h}}{\delta ^{\beta _-}} =0.In simple local super-harmonic functions \overline{H} and \overline{h} have been constructed with the property that\lim _{d(x,\partial \Omega )\rightarrow 0} \frac{\overli... | {
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potential | [
"Catherine Bandle",
"Maria Assunta Pozio"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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b7647b9497cda0bfbdbbb0a75b85407fb7c33d86 | abstract | 0 | 4 | Abstract | Dependently typed languages are well known for having a problem with code
reuse. Traditional non-indexed algebraic datatypes (e.g. lists) appear
alongside a plethora of indexed variations (e.g. vectors). Functions are often
rewritten for both non-indexed and indexed versions of essentially the same
datatype, which is a... | {
"cite_spans": []
} | 1803.08150 | Generic Zero-Cost Reuse for Dependent Types | [
"Larry Diehl",
"Denis Firsov",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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40fd448f2666a1b3e11f01117543ae1d360cb87f | subsection | 1 | 4 | Introduction | Dependently typed languages
(such as Agda , Coq ,
Idris , or Lean ) can be used to
define ordinary algebraic datatypes, as well as indexed versions of
algebraic datatypes that enforce various correctness properties.
For example, we can index lists by natural numbers to
enforce that they have a particular length
(i.e. \... | {
"cite_spans": []
} | 1803.08150 | Generic Zero-Cost Reuse for Dependent Types | [
"Larry Diehl",
"Denis Firsov",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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18e52729e256c0615707efbfd0921748be9427e0 | subsection | 2 | 4 | Introduction | Section REF : A generic combinator solution to zero-cost
forgetful data reuse
(combinator ifix2fix, handling the type of fixpoints for
generically encoded datatypes).
Section REF & Section REF : Generic combinator solutions to zero-cost
enriching data reuse
(combinators fix2ifix and fix2ifixP, handling the type of fix... | {
"cite_spans": []
} | 1803.08150 | Generic Zero-Cost Reuse for Dependent Types | [
"Larry Diehl",
"Denis Firsov",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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1133a8bd33dbbf440f504ef6a4013ce0ef7e90fa | subsection | 3 | 4 | The Type Theory (CDLE) | We briefly summarize the type theory, the Calculus of Lambda
Eliminations (CDLE), that the results of this paper depend on. For
full details on CDLE, including semantics and soundness results,
please see the previous papers , , . The main
metatheoretic property proved in the previous work is logical
consistency: there ... | {
"cite_spans": []
} | 1803.08150 | Generic Zero-Cost Reuse for Dependent Types | [
"Larry Diehl",
"Denis Firsov",
"Aaron Stump"
] | [
"cs.PL"
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7c1aac5b89646ea1136d855c71aa54f5eae44784 | abstract | 0 | 77 | Abstract | We establish sufficient conditions for exponential convergence to a unique
quasi-stationary distribution in the total variation norm. These conditions
also ensure the existence and exponential ergodicity of the Q-process, the
process conditionned upon never being absorbed. The technique relies on a
coupling procedure t... | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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ee7d1af01198e5bcc06bbc13f2d36e9cc162fbaf | subsection | 1 | 77 | Presentation | Given a continuous-time and continuous-space
Markov process with an absorbing state,
we are interested in this work
in the long time behavior of the process
conditionally on not being absorbed
–not being "extinct".
More precisely,
we wish first to highlight key conditions on the process
such that the marginal of the pr... | {
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"... | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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b4d1f3b63fa1277d275e2f9c3b5d8fde70a60a1a | subsection | 2 | 77 | Presentation | Once all the conditions are met,
Harris recurrence ensures the existence of a measure minorizing the DCNEs
and whose mass tends to one,
thus ensuring contraction in total variation norm.Given the vast literature on QSDs
(see notably the impressive bibliography collected by Pollett ),
it is difficult to assert
that no o... | {
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"raw": "Pollett, P. K.; Quasi-stationary distributions: A bibliography., available at people.smp.uq.edu.au/PhilipPollett/papers/qsds/qsds.html (2015)",
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extinction | [
"Aurélien Velleret"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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b128fc2fb408dff5fe76ca816c4fb066b4b51000 | subsection | 3 | 77 | Presentation | To our knowledge, we are only preceded by a few months
by Champagnat and Villemonais in .
They also manage to obtain such kind of convergence,
even beyond the case of a unique QSD.
Like for us,
the criteria they introduce
are analogous to older techniques involving,
in the case of discrete-time and discrete-space proce... | {
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"... | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
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ec6feaf9cbe8bdec9eda40de3bd68e5e32b9edcb | subsection | 4 | 77 | Presentation | We then explain why we can simply consider
the first from our two sets of assumptions
in order to complete the proofs of the three theorems.
These proofs are finally given in Section .
In Section ,
we present two applications of our general theorems.
Theses results seem to be new,
but concern toy-models.
We hope that t... | {
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} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
"math.PR"
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fdc7590d27d953ba698c11c732d19c2b2b313e95 | subsection | 5 | 77 | Elementary notations | In the following, the notation k\ge 1 has generally to be understood as k\in while \ge 0 –resp. c>0– should be understood as \in _+:= [0, \infty ) –resp.
c\in _+^* := (0, \infty ). In this context (with m\le n),
we denote classical sets of integers by :
\quad _+~:= 0,1,2...,\; ~:= 1,2, 3...,
\; [\![m, n ]\!]~:= m,\, m+... | {
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extinction | [
"Aurélien Velleret"
] | [
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bb653b14d7186cef41253c49ce416464d40432e7 | subsection | 6 | 77 | The stochastic process with absorption | We consider a strong
Markov processes absorbed at \partial : the cemetery.
More precisely, we assume that
X_s = \partial implies X_= \partial for all \ge s.
This implies that the extinction time :
\quad ~:= \inf \ge 0 X_= \partial \quad
is a stopping time.
Thus, the family (P_)_{\ge 0}
defines a non-conservative semi... | {
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"doi": "10.1214/11-ps191",
"end": 1636,
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"source_ref_... | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
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c80a6c1f7ab125c5b4769c90077309c50697d8ea | subsection | 7 | 77 | Specification on the state space | In the following Theorems, we will always assume :
-1
:
"Exhaustion of "There exists a sequence (_n)_{n\ge 1} of closed subsets of such that :{n\ge 1} _n \subset _{n+1} ^\circ \quad \text{ and } \quad _{n\ge 1} _n= .This sequence will serve as a reference for the following statements.
For instance, we will have control... | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
"math.PR"
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0d983c3582027aa739fe68edb1e759187639f309 | subsection | 8 | 77 | Hypotheses | Our results rely on a set \mathbf {(A)} of assumptions
which is actually implied by a much stronger
yet simpler set of assumption \mathbf {(A^{\prime })}.
We detail first each basic assumption
and then define \mathbf {(A)} and \mathbf {(A^{\prime })}
in terms of those.We recall that for any set ,
we defined the exit an... | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
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0.00... |
8e3384e09ce4f3c9a688c0ed42196f5ac23fc9f5 | subsection | 9 | 77 | Hypotheses | Moreover, there exist >
and a given set _c \in for which
[, _c] and [,_c] hold."Given stronger versions of and ,
we no longer need any "survival estimate",
so that we can simplify the previous assumption:We say that the (stronger) assumption \mathbf {(A^{\prime })} holds, whenever :", and [] hold, with some \in .
... | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
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6b92a2bf1f20bcf9ce453963b511fbf512544396 | subsection | 10 | 77 | Main Theorems : the simplest set of assumptions | Assume that \mathbf {(A)} or \mathbf {(A^{\prime })} holds.
Then, there exists a unique QSD \alpha .
Moreover, we have exponential convergence to \alpha of the DCNE's
at a given rate.
More precisely, there exists \zeta >0
and to any pair \ge 0 and \xi >0,
we can associate a constant C(,\, \xi ) such that :{> 0}
{\mu \i... | {
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"... | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
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1d10ab55c19f357a2c8379e37032596b10b22a3c | subsection | 11 | 77 | Main Theorems : the simplest set of assumptions | It also belongs to the domain of the infinitesimal generator , associated with the semi-group (P_)_{\ge 0} on (B(\cup \lbrace \partial \rbrace ); {.}), and :\, \eta = -\, \eta , \qquad so\quad {\ge 0}
P_\, \eta = e^{-\, } \eta .
{EFeta}{}Remark :
As in ,
it is also not difficult to show that there is no eigenvalue of ... | {
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"source_ref_id": "b0b227641a... | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
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] | [
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91c43aacb757c12384092747f9f13c4019458578 | subsection | 12 | 77 | Main Theorems : the simplest set of assumptions | In other words, for all \varphi \in _b() and \ge 0,
\quad \langle \delta _x\, Q_\varphi \rangle = e^{\, }\,
\langle \delta _x \,P_\eta \times \varphi \rangle \, / \,\eta (x), \quad
where (Q_)_{\ge 0} is the semi-group of X under .(iii) Exponential ergodicity :There is a unique invariant distribution
of X under ,
defin... | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
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ed01e04e1dcfa6bc82862f46a9eda2b8e1f3a084 | subsection | 13 | 77 | How to verify ? | For discrete space, it is quite natural to deduce
from the fact that there exists s.t. :
\quad \inf _{x\in _c} _{}(X_= x) > 0.\quad
We can then couple trajectories from and from x
with a time-shift of length
(see the birth and death process for an illustration).For continuous space, we need however to wait a bit fo... | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
"math.PR"
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1bb8f05a0be1d1ca38d2074064463ff977e59da8 | subsection | 14 | 77 | Further comparison with the literature | To our knowledge,
the kind of dependency
on the initial condition \mu
that we propose for the convergence
has not previously been introduced,
except as a specific case of the very new result of .
It is however quite natural
for the models we have in mind,
where extinction plays a stabilizing role,
preventing transient... | {
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... | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
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92087864bd7f9e69119c9b605dd4f1652119a84d | subsection | 15 | 77 | Remarks on the Assumptions | The counter after each set of Remarks refers to the related assumption.Remark REF :
For any n \ge 1 and any initial condition, the exit time T_{_n} and the hitting time \tau _{_n} are stopping times, as well as any iterated combination of the kind "first hitting time of _n after the exit time of _m" –cf. Theorem 52 in... | {
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extinction | [
"Aurélien Velleret"
] | [
"math.PR"
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18b433ed12dccea4288477b8d0dc6122b13b9c6d | subsection | 16 | 77 | Remarks on the Assumptions | In any case, we only need to control the process on some finite time-interval to state Rg.{1} Relations between the sets of assumptionsThe following Lemmas notably prove the implication
\mathbf {(A^{\prime })} \mathbf {(A)},
and more generally indicates a way to prove .
We also obtain a visibly stronger yet equivalent ... | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
"math.PR"
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98c499e145b50f70039ca874d955bcde7a20df50 | subsection | 17 | 77 | Regeneration estimate : | : "From mixing to regeneration"
Assume that there exists \in and _s\subset _{}
such that _s > 0 and holds.
Then, there exists ,\,> 0 and _{^{\prime }}\in s. t. :&{x\in _s}
\hspace{14.22636pt}
_{x} X_{} \in _s< {\le } X_\in _{^{\prime }} \ge .
{Rg}{R_g}: "From regeneration to survival"Assume that there exists t_{rg},... | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
"math.PR"
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f4876ebf0393f24f5b372790cbdbfbbc7480d474 | subsection | 18 | 77 | Proof that | By the two previous lemmas,
we know that in \mathbf {(A^{\prime })},
we can include w.l.o.g.
for some >0 and _s\in
s.t. (_s)>0.
Then we can define (arbitrarily) some >
and deduce the existence of _c for which and hold.
This concludes that \mathbf {(A^{\prime })} is indeed stronger than \mathbf {(A)}. | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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c1abe61ed124525def2b61b07d18026f1e98b9ea | subsection | 19 | 77 | Proof of Lemma | To produce a regeneration estimate, we apply with
n_{mx} = .
Then, we obtain Rg, where we
define t_{rg} the associated value of , ^{\prime }_{rg} the associated value of m_{mx}, c_{rg}~:= \; (_s) >0.Assume Rg. Let x \in _s,
~:= -{t_{rg}} \ln (c_{rg}),T_{}:=\inf \ge 0,\, X_\notin _{} . | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
"math.PR"
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d8b0071f9402dab048d771e72ddeb1f43ce82036 | subsection | 20 | 77 | Proof of Lemma | By induction over k \in and the Markov property :{k\ge 1}\qquad \inf _{x\in _s}
_{x} ( k \, t_{rg} < T_{} )
\ge \exp (- \,k \, t_{rg}).Thus, for a general value of t>0 :&\inf _{x\in _s} _{x} ( < T_{} )
\ge \inf _{x\in _s}
_{x} \left\lceil \tfrac{}{t_{rg}} \right\rceil t_{rg} < T_{} \ge \exp - \,\left\lceil \frac{}{t_{r... | {
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} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
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ffa140863b32562669205b28e9a96bb9dd444810 | subsection | 21 | 77 | Proof of Lemma | \quad {equ}Let n\ge 1, \xi >0.
By , we can thus define c_m > 0t_m > 0 \in such that _s > 0 and :
{\mu \in }\quad _\mu {X}_{t_m}\in _s
\ge \xi \, c_m\, ( _s).
We then apply with the Markov property :
{\mu \in }{t_e>0}&_\mu t_e < \ge \xi \, c_m\, ( _s) \times c_{sv} \,e^{\, t_m}\, e^{-\, t_e}.
{eqt}Thus, by and , wi... | {
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extinction | [
"Aurélien Velleret"
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06f18aa7580a7fbf9f875a15b62b36e11e28601d | subsection | 22 | 77 | Proof of Lemma | Since adaptation is poor outside _{bk}, it is not much more "costly" to impose that the process gets inside and then stay confined in _{out}.Thanks to and since _s \subset _{bk},
we know that there exists some , t_{bk}, c_{bk} > 0 such that,with still T_{xt}~:= \inf s> 0X_s \notin :&\hspace{14.22636pt}{x \in _{\ell ... | {
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extinction | [
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b02b46398cc79d1b0fc3a2d91a955e0bdd76451d | subsection | 23 | 77 | Proof of Lemma | Indeed, compared to trajectories that stay –almost– inside
(in particular those reaching quickly _s and not leaving _m)
they vanish in probability with a larger rate : \rho _{eT} > .
In practice, we want to do this comparison at the time the process
exits _{bk} for the last time before t_e : T^{I(t_e)}_{out}.
Yet, it i... | {
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} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
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1f151255ae03b174b8c852addc961fcf09ff4cb2 | subsection | 24 | 77 | Proof of Lemma | Finally~:
\begin{align*}
&P^{\ref {TteOut}} \le ( e_ c_{xt})e^{- (- ) \, t_m} {i\ge 1}
_\mu T_{out}^i \le t_e - t_m t_e < T_{out}^{i+1}\wedge \\
&\hspace{28.45274pt}
\le ( e_ c_{xt})e^{- (- ) \, t_m} _\mu t_e < ,
\quad \text{ since the sets are disjoint,}
\end{align*}
which concludes the proof of Lemma \ref {TteOut} wi... | {
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} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
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19e9ccffab11ac1a13a7d3f2babf25b7466987a3 | subsection | 25 | 77 | An equivalent version of | is equivalent to the apparently stronger version :&\hspace{14.22636pt}
{\ge 1}
{t_\veebar \ge 0}
\hspace{14.22636pt}
{> }
{\ge t_\veebar }
{>0}
\\
&\hspace{28.45274pt}
{x \in _{}}
\hspace{14.22636pt}
_x {X}_{}\in dx < {\le } X_\in _{} \ge \, (dx),with the same measure –so that the condition (_s)>0 is preserved.Proof o... | {
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extinction | [
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98e6f1aaea169df7fa6b4494735180519c00b974 | subsection | 26 | 77 | Birth-and-death process with catastrophes | We choose to illustrate our result with this example
for its clear simplicity.
We have to admit though that it seems to be also easily treated
by the new criteria of (cf. Remark (vi)).To ensure the uniqueness,
we will impose that
the catastrophe rate is large enough when the population size is large.
Biologically, we c... | {
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d74c6ced25c2729b10f3775705cb23d49bc8ab25 | subsection | 27 | 77 | Description of the process | X, the population size,
is a time-homogeneous Markov Chain on _+
where \partial = 0 the absorbing state and = .
Given X_0 = n\ge 1, there is
a death with rate d_n>0 (leading to X = n-1),
a birth with rate b_n>0 (leading to X = n+1)
and a catastrophe with rate c_n\ge 0 (leading to X = 0).Assume that : \quad
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c22b4bf4413c48723f80ec321fafd05a46baab10 | subsection | 28 | 77 | Description of the process | Then can easily be adapted with K \in _c = [\![1, n_c ]\!].
The proof of the other assumptions remain the same.Our Theorem REF could also be proved
using Theorem 5.1 in
with the Lyapunov function
\psi _1 \equiv {}.
Their is clearly upper-bounded by \inf _{k\ge 1} (b_k + d_k + c_k)
–this is the intuition of .
We coul... | {
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e0ca8097606e8be5f7bbd6896533b62d9d0f9055 | subsection | 29 | 77 | Proof of Theorem | Proof of the exponential convergenceBy 0, let k, n_c\ge 1 and >0 be such that :&0 < ~:= b_k + d_k + c_k
\,<\, \,<\, \textstyle {\inf _{\lbrace n \ge n_c +1\rbrace }} c_n
:= {\rho }_{eT}
{RhoBD}{BD}
\\
&\hspace{28.45274pt}
~:= \delta _k,\quad _s~:= k , \quad _c~:= [\![1, n_c ]\!]Let _n = [\![1, n\vee k]\!], for n\ge 1.
... | {
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93cf538152f6556ba254e58ad3d018a8a2545edc | subsection | 30 | 77 | Proof of Theorem | We can indeed choose := \delta _k, m_{mx} := n
and the value of c_Y
associated to an arbitrary choice of t_Y = t_{abs}.With i := k and n = n_c,
0
and the Markov property implies :{t>0}
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25fc4294fd3884af6028d8711c4c1d35f1e5eed1 | subsection | 31 | 77 | Proof of Theorem | Thus, we won^{\prime }t detail it much and refer to \cite {ChpLyap}.Let \in _1(), \ge ~:= \vee and x\in .&_x(< )
\le \, _x_{} (t- < {}) < (t-)\wedge + _x (t-\le \wedge ) {equ}thanks to property , since t-\ge \ge on
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2e6b083bfea40f1a222b88b6fa5372983c751a9f | subsection | 32 | 77 | Proof of Theorem | \\&\hspace{56.9055pt}
\text{Thus }
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eb90e6316f164186fee1bf7dd739cc61ab5a07f2 | subsection | 33 | 77 | Proof of Theorem | For t sufficiently large (a priori depending on \mu ), we deduce from
, etaInf and ECvEta :
\qquad 0 \le M_s^t
\le 2\, e^{s}\,{\eta _\bullet }\ /\ \langle \mu \eta \rangle .\quadThus, by the dominated convergence Theorem, we obtain that
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659be92fbc4706b00f5063a435cdbc5c97e79bc0 | subsection | 34 | 77 | Proof of Theorem | By the previous calculations, and etaInf, for any >0 :&
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\\&\hspace{28.45274pt}
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1ae6929556cea8c2b1a5c9a229b7344de440cbec | subsection | 35 | 77 | Proof of Theorem | We prove uniqueness with the next subsection.{1} Step 3 : Proof of ECvBeta :It is relatively easy to adapt the proof of Theorem REF
for the case of the Q-process
by generalizing Proposition REF into :
For J\, t_{db }\le \le ,
let :&\alpha _{c~: J}^{} (,\, dx)
:= {k \le J} ()\, 1-^{k-1}
\, _{}X[- k\, ] \in dx - k\, <... | {
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61ade92330aa2f292a6269dd11fe46075b8a3203 | subsection | 36 | 77 | Proof of Theorem | Then :&{t\in [J\, , ]}\qquad _{\mu }X_{}\in dx <\ge \alpha _{c~: J}^{} (,\, dx).The proof Lemma REF
is easily adapted from the proof of Lemma REF
once one remarks :&_{\mu }X_{}\in dx^{\prime } <= \dfrac{_\mu (J\, < )}{_\mu (< )} \; _{x^{\prime }}(-< )\;
\mu A_{ J\, } P_{- J\, }(dx^{\prime }),
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71d0b07b7c0ce87c9e29d28f04d981d665d185d7 | subsection | 37 | 77 | Proof of Proposition | Proof of the non-uniformityWe consider one of the simplest choice,
which is to take b_n, d_n linear in n (the classical Malthus' growth model,
without competition)
and c_n constant for n\ge 2.
We can then choose arbitrarily :&b_1, c_1, \bar{b}, \bar{d} \in (0, \infty )^5, \quad c_2 > (b_1 + c_1),
{Bdc}{b,d,c}\\
&d_1 = ... | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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6f3ec50e584b871c4efafffbcdcb3594c3946005 | subsection | 38 | 77 | Proof of Proposition | \\&_x ( \tau _N \le < ) \le e^{-c_1\, }\, _x ( \tau _N \le )
\le \; e^{-c_2\,} / 2.Since the extinction rate is upper-bounded by c_2 :
_x ( < )
\ge e^{-c_2\, } _x ( \tau _N \le < ) \le /2.
Therefore, with also :&\Vert \delta _1 A_- \delta _x A_\Vert _{TV}
\ge _1 (X_\le 2^N < ) - _x (X_\le 2^N < ) &
\\&{1}
\ge 1- /2 - ... | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
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] | 2,018 | en | Mathematics | [
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222da58ec41b1dbe4b194d08ea52cfaa0132449c | subsection | 39 | 77 | Adaptation of a population to its environment : application to a diffusion process | In this illustration,
the notion of being in a mal-adapted region
is quite intuitive
and the criteria for the exponential convergence
to a unique QSD rather natural.
Again, the general proof
for this illustrative example
is unclear without our techniques,
except maybe with those of .
Yet, in this case,
it is presumably... | {
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"doi": "10.1214/22-ejp880",
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... | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
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0b84f54a074f4c37388d9ce38ac76b4cb012d44d | subsection | 40 | 77 | Presentation of the model | We consider a simple coupled process describing
the eco-evolutive dynamics of a population.
We model the population size by a logistic Feller diffusion (N_t)_{t\ge 0}
where the growth rate (r(X_t))_{t\ge 0}
is changing randomly.
Namely,
the adaptation of the population
and the change of the environment
are assumed to a... | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
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fa6b3029b53892923d2edd05664a0c9f51444321 | subsection | 41 | 77 | A precise theorem for the strong Assumption | In the following, we say that a process (Y_t) on
with generator
(including possibly an extinction rate \rho _c)
satisfies Assumption (H) whenever :For any compact sets K, K^{\prime }\subset
with C^2 boundaries s.t. K\subset int(K^{\prime }),
0<t_1<t_2 and positive C^\infty constraints :
u_{\partial K^{\prime }} : (\... | {
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"doi": "10.1007/bfb0082920",
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"source_ref_id": "2122d67d19... | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
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4c8a4d4fd1ba1a0486a6c9e9b372763c1365ee63 | subsection | 42 | 77 | A precise theorem for the strong Assumption | We describe the solutions of (S)
rather to give an intuition on the parameters
for the survival estimate.(ii) The proof would also still hold
if in Assumption (H),
t_2 happens to depend on K and K^{\prime }.(iii) A priori, the estimates given by the Harnack inequality
are usually rough.
For more precise estimates,
we w... | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
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3374542bf257f192d75f246c0f71ce465c3b52da | subsection | 43 | 77 | and are implied by the Harnack inequality | {1} Assumption (H)
with Y_t = (X_t, N_t)
implies :We define _n a sequence of strictly increasing compact sets
with C^2 boundaries whose union is := ^d \times _+^*.
For some C^\infty function f with support in _n = K,
we apply Assumption (H) with
u_{\partial K^{\prime }}(t, y) := f(y) on \lbrace 0\rbrace \times _n
and u... | {
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extinction | [
"Aurélien Velleret"
] | [
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d6b9554fa0d2142ebb29fce373edd34194c7573d | subsection | 44 | 77 | and are implied by the Harnack inequality | Thus, we approximate it
on the parabolic boundary [t_,\, \infty ) \times \partial \,
\bigcup \, \lbrace t_\rbrace \times
by the family (U_k)_{k\ge 1} of smooth –_+^\infty w.l.o.g.– functions.
We then deduce approximations of u in [t_,\, \infty ) \times
by (smooth) solutions of :&\hspace{28.45274pt}
\partial _u_k (, y... | {
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extinction | [
"Aurélien Velleret"
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ab41075987d8d8e731cd40514e3fbc8076d719a9 | subsection | 45 | 77 | Discussion about the survival estimate | The issue that we discuss here
is the way to deduce
a (not too rough) lower-bound on the survival rate.
In practice, it is given
for an initial condition in an interior subspace,
where the process is killed when it exits some compact set _m.
For simplicity, assume here for instance that the demographical dynamics (on N... | {
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extinction | [
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aa4d37f8d5da848da16c0e548f0fafa5112ee36f | subsection | 46 | 77 | Discussion about the survival estimate | In any case,
as long as ETDif holds for some \rho > ,
our argument will deal with the transitory domain
without further condition.In the following,
we propose to show how to get some (rough) estimate
of the form ETDif for any \rho
by taking _c sufficiently large,
thus concluding Assumption \mathbf {(A)}
and ensuring T... | {
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} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
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f11e151e7af88a07ad85d45a60b8b508fb4a1d04 | subsection | 47 | 77 | Proof of for any | {1} Decomposition of the transitory domainThe complementary of c is then made up of 3 subdomains :
"y=", "y=0", and "x = ", according to figure \ref {eTD}.
Thus, we define:
\begin{}
\item :=
^d \setminus B(0n_c) \times (y_{\infty }, \infty )
\bigcup \; B(0n_c)\times (n_c, \infty )\hfill ("y= \infty ")
\item 0 := B(0n_c... | {
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} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
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576d4816d0d39b29bd761b1602b0164bc3f8c40c | subsection | 48 | 77 | Proof of for any | We refer to Appendices A, B and C
for the (technical) proofs of the propositions (including the lemmas),
but show at the end of this Subsection REF
how to deduce from the three propositions that follow :Given any \rho >0,
we can define y_{\infty } > 0 and C_{\infty }^Y \ge 1 s.t.,
whatever n_c> y_{\infty } :^Y_\inft... | {
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} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
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440ef30bdd7c7416209248395ff77f970233a7e1 | subsection | 49 | 77 | Proof of for any | Then, whatever the time t_D >0 (that we choose for the descent)
and the error >0,
we can find a lower-limit >0 of –sufficently large–,
such that ::=
\inf t\ge 0, \; Y^D_t \le ,
{ y > 0} {1}
_{y} (t_D < )
\leProposition REF relies on the strong negativity on the drift term :
Considering any c_Y, t_D >0, with
^D := \in... | {
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extinction | [
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1e9113c70cf6323f7f42581ef3f6c982d326b04e | subsection | 50 | 77 | Proof of for any | On the other hand,
the restrictions on the transitions are crucial,
since otherwise, for instance with a limit-cycle,
the process could persist by circulating between these areas.{1} Combine all the inequalities to proveWe will first prove that
the inequalities EYi, EXi and EY0
give an upper-bound of the global supre... | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
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8b05520736a20f6dddc6a3b70537147be547f560 | subsection | 51 | 77 | Proof of for any | To make the inequalities EXi and EY0 hold,
we can just take n_c := n_c^X \vee n_c^0.Remark :
What is essential for this proof
is the fact that,
when all the "exceptional transitions" have been neglected,
the graph of transitions
has no loop.
Then, we have to justify
that we can choose the different values for
such ... | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
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5b945c0fd8489ef3052490dedadec0ee54051aef | subsection | 52 | 77 | Proof of Theorems | In Subsection REF , we present
the general principles of our coupling
that concludes the proof of Theorem REF .
These principles would alone end the proof
in the context of the Assumption (A) in .
Yet, with our more general assumptions,
they rely on the results of the two previous subsections.
First, we prove in Subsec... | {
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898437f6b18362fe592f2d04b8c64e8a5de05c51 | subsection | 53 | 77 | Stabilization of the process in the long run | The main purpose of this section is to prove :Assume that \mathbf {(A)} holds { plays no role here}.
Then, there exists _{xt} = _{n_{xt},\, \xi _{xt}}
(with n_{xt} \ge m_{sv}, \xi _{xt} > 0) and,
to any pair \ge 0 and \xi >0,
we can associate a time
t_{xt} = t_{xt}(n, \xi )>0 such that :&{\mu \in }
{\ge }\qquad \mu A_... | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
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c40bc8e0cb49c41da2b0e2f0fa3a755b016befa8 | subsection | 54 | 77 | Stabilization of the process in the long run | Then,
there exists n_{xt} > n_{\ell j},
~:= _{n_{xt}}
and c_{xt} > 0 s.t.
with T_{xt}~:= \inf s> 0X_s \notin :&{x\in _{\ell j}}
{t > 0}
\hspace{14.22636pt}
_{x} < T_{out}^1\wedge T_{xt}\wedge \ge c_{xt}\, \exp [- \, t].: Last exit from _{out}Suppose that , , , 0, and eqlj hold,
with > ,
_s \subset _{m} \subset _{bk}... | {
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extinction | [
"Aurélien Velleret"
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0.013959524221718311,
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... |
ac7dd94aeb495b38b1e23554a0f1c7c133a92385 | subsection | 55 | 77 | Proof that Lemmas | Proof of 0With Lemma REF and REF , we obtain an upper-bound –with high probability– on how much time the process may have spent outside _{out}.
Thus, we can associate most of trajectories ending outside _{out}
to others ending inside _{out}. From this association, we deduce a lower-bound on the probability to see the p... | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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7cdf9d916b162573885c87bf22cdbd936a400d90 | subsection | 56 | 77 | Proof that Lemmas | Since a.s.\; I(t_e) < \infty (Lemma REF ) :&_\mu t_e - t_m < T_{out}^{I(t_e)}\tau _{bk}^1 \le t_et_e < \\&\hspace{19.91684pt}
= {i\ge 1}
_\mu _{X_{ T_{out}^i}} t_e - T_{out}^i < {}\wedge {T_{out}^{1}} t_e - t_m < T_{out}^i \\&\hspace{28.45274pt}
\le {i\ge 1}
_{\mu } t_e - t_m < T_{out}^i \\&\hspace{14.22636pt}
\le (e^{... | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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d7d45bd065cf4d635675cb1811df122dd7975340 | subsection | 57 | 77 | Theorem | For the proof of the following Theorem REF ,
we need the following Corollary of Theorem REF :
"Stability" :Assume \mathbf {(A)}. Then,
there exists , >0 such that :&{ u\ge 0}
{\ge u + }
\hspace{14.22636pt}
P_{}(- u < )
\le \; e^{\, u}\; _{}< .
{Sb}{Sb}Assume that there exists \rho _{eT} \ge >0, _s \subset , _c \subse... | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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... |
84cb580f7fcb515de5a64e3071d8ed2846177879 | subsection | 58 | 77 | Proof of Corollary | Proof of 0:Since this proof is elementary, we will not go into details :Under some condition t-u\ge t_{sb} that comes from :&_{}< \ge c_{sv} \,\exp (-\, u) \, _{}t-u < A_{t-u}(_s)
,
\\&{1}
\ge c_{sv} \,\, \, \exp (-\, u) \, _{}t-u < {0}+. | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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45f0a365b1af47c7a8b1fd64188f7f33d7dbd167 | subsection | 59 | 77 | Definition of the expected uncoupled part | With a given set of parameters ,\, ,\, ,\, >0
–cf. following subsection– we define\text{for } > ~:\quad J()
~:= (- )/.
{Jtobs}{J()}For \ge 0, \mu \in ,
> , and k\in ,
let :&(k, t) = a_{\mu }^{}(k, t)
~:= {k\le J(),\, k\, \le }
\times \, ( 1-)^{k-1}
\\&\hspace{113.81102pt}
\times \dfrac{_\mu (< )}{_\mu (< )}
\times \dfr... | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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0.0... |
d989b56fd6ff26eb03857952d5cb2e1deda316e1 | subsection | 60 | 77 | Definition of the constants involved | For clarity, we denote by (for horizon of time) the time t that appears in Theorem REF . During this coupling procedure, it will stay fixed, and won't appear in the other sections.
The constants c_{ps}, t_{ps}>0 come from Theorem REF ,
while c_{db}, t_{db}>0 come from this corollary of Theorem REF :
"Coupling and Ren... | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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0.... |
72ce585a4a57e88e168327b01ee4540d7aaeee2c | subsection | 61 | 77 | Definition of the constants involved | Subsection REF – such that :&{x\in }
_x X_{} \in dx< \ge \; (dx).
{eqd}&\hspace{14.22636pt}\text{With }
~:= \; \xi _{rn}\in (0,1),\; ~:= \ge t_{rn},
\text{ we deduce : }\\
&{\mu \in _{rn} }
\mu A_{}(dx)
\ge \mu ()\, \, (dx)
\,\\
&\hspace{71.13188pt}
\ge \xi _{rn}\, \, (dx)
= \; (dx)
\qquad \text{ because } \mu \in _{rn... | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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0... |
e1e2ecdb2dea477f83b17c15901f47efe73cdd7c | subsection | 62 | 77 | Lower-bound on the marginals | At time t_h,
for any initial condition \mu \in _{rn},
the DCNE shall be lower-bounded by :[](dx)
&= (;\,dx)
~:= {k \le J()} ()( 1-)^{k-1}
\, A_{-k\, }(dx) \ge 0
{eqc}{[] }Remark :
The definition of ([t])_{\ge 0} implicitly depends on , , and , but not on \mu , n or \xi .The proof of Theorem REF will be completed thank... | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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0... |
ca0d69b5a57416b1ff5b4054a277af195f53cbaf | subsection | 63 | 77 | Lower-bound on the marginals | By letting ^2 go to infinity
in the last inequality,
we further get an exponential rate of convergence \zeta ,
independent of n and \xi ,
and in particular on \mu .Since immediately : \quad {n \le n^{\prime }}
{ \xi \ge \xi ^{\prime }> 0}
\quad \subset _{n^{\prime },\, \xi ^{\prime }},\quad
we deduce \alpha ^{n,\, \xi... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1214/09-aop451",
"end": 815,
"openalex_id": "https://openalex.org/W2951886511",
"raw": "Cattiaux, P., and all; Quasi-Stationary Distributions and Diffusion Models in Population Dynamics, The Annals of Probab., V. 37, No. 5, pp. 1926–1969... | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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... |
0a1c7bd7f1abee5017088adb05ba10cef1729f6e | subsection | 64 | 77 | Lower-bound on the marginals | Then \mu A_{} \ge [].{1}
Proof of Proposition REF
with Lemmas REF , REF and REFLet us first assume that \mu \in _{rn},
where we use Proposition REF together with and 0 to define
_{rn} such that Mrn holds.Then, by induction over j\le J(), we state
(I_j)~:~r_{j} > 0\text{ and }\nu _j \in _{rn}.\
We initialize at j ... | {
"cite_spans": []
} | 10.1016/j.spa.2022.02.004 | 1802.02409 | Unique Quasi-Stationary Distribution, with a possibly stabilizing
extinction | [
"Aurélien Velleret"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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... |
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