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Theorem 67.9.5 Let \( M \) be an \( {\mathcal{G}}_{t} \) martingale and suppose \( M\left( t\right) \in {L}^{2}\left( \Omega \right) \) for all \( t \geq 0 \) . Then there exists a unique stochastic process, \( \mathbf{g}\left( {s,\omega }\right) \) such that \( \mathbf{g} \) is \( {\mathcal{G}}_{t} \) adapted and in \... | Proof: First suppose \( \mathbf{f} \) is an adapted function of the sort that \( \mathbf{g} \) is. Then the following claim is the first step in the proof. Claim: Let \( {t}_{1} < {t}_{2} \) . Then \[ E\left( {{\int }_{{t}_{1}}^{{t}_{2}}{\mathbf{f}}^{T}d\mathbf{W} \mid {\mathcal{G}}_{{t}_{1}}}\right) = 0 \] Proof of cl... | Yes |
Let \( H = {L}^{2}\left( \left\lbrack {0, T}\right\rbrack \right) \). For \( f \in H \), let\n\n\[ W\left( f\right) \equiv {\int }_{0}^{T}f\left( u\right) {dW} \]\n\nwhere \( W\left( t\right) \) is the one dimensional Wiener process. | First of all, note that the integrand is adapted to the usual filtration determined by the Wiener process. This is because \( f \) does not depend on \( \omega \) . That \( W\left( f\right) \) is normally distributed can be seen from the approximation of the Ito integral with the integral of elementary functions. These... | Yes |
Example 68.0.3 You could let \( H = \mathbb{R} \) and let \( \xi \) be normally distributed density with mean 0 and variance 1. Then let \( W\left( a\right) \equiv {\xi a} \) . Does it work? | \[ E\left( {{ab}{\xi }^{2}}\right) = {abE}\left( {\xi }^{2}\right) = {ab} \] Is \( W\left( a\right) \) normally distributed? \( E\left( {e}^{ia\xi t}\right) = {\int }_{\mathbb{R}}{e}^{iaxt}{\xi dx} = {e}^{-\left( {1/2}\right) {a}^{2}{t}^{2}} \) which is the characteristic function of a normally distributed random varia... | Yes |
Theorem 68.1.3 The Hermite polynomials are the coefficients of a certain power series. Specifically,\n\n\\[ \exp \\left( {{tx} - \\frac{1}{2}{t}^{2}\\lambda }\\right) = \\mathop{\\sum }\\limits_{{n = 0}}^{\\infty }{H}_{n}\\left( {x,\\lambda }\\right) {t}^{n} \\] | Proof: Replace \\( {H}_{n} \\) with \\( {K}_{n} \\) which really are the coefficients of the power series and then show \\( {K}_{n} = {H}_{n} \\) . Thus\n\n\\[ \exp \\left( {{tx} - \\frac{1}{2}{t}^{2}\\lambda }\\right) = \\mathop{\\sum }\\limits_{{n = 0}}^{\\infty }{K}_{n}\\left( {x,\\lambda }\\right) {t}^{n} \\]\n\nTh... | Yes |
Theorem 68.2.3 \( {L}^{2}\left( {\Omega ,\mathcal{F}, P}\right) = { \oplus }_{n = 0}^{\infty }{\mathcal{H}}_{n} \) . The symbol denotes the infinite orthogonal sum of the closed subspaces \( {\mathcal{H}}_{n} \) . That is, if \( f \in {L}^{2}\left( \Omega \right) \), there exists \( {f}_{n} \in {\mathcal{H}}_{n} \) and... | Proof: Clearly each \( {\mathcal{H}}_{n} \) is a closed subspace. Also, if \( f \in {\mathcal{H}}_{n} \) and \( g \in {\mathcal{H}}_{m} \) for \( n \neq m \), what about \( {\left( f, g\right) }_{{L}^{2}\left( \Omega \right) } \) ?\n\n\[{\left( f, g\right) }_{{L}^{2}\left( \Omega \right) } = \mathop{\lim }\limits_{{l \... | Yes |
Corollary 68.2.4 Let \( {\mathcal{P}}_{n}^{0} \) denote all polynomials of the form\n\n\[ p\left( {W\left( {h}_{1}\right) ,\cdots, W\left( {h}_{k}\right) }\right) ,\;\text{ degree of }p \leq n,\text{ some }{h}_{1},\cdots ,{h}_{k} \]\n\nAlso let \( {\mathcal{P}}_{n} \) denote the closure in \( {L}^{2}\left( {\Omega ,\ma... | Proof: It is obvious that \( {\mathcal{P}}_{n} \supseteq { \oplus }_{i = 0}^{n}{\mathcal{H}}_{i} \) because the thing on the right is just the closure of a set of polynomials of degree no more than \( n \), a possibly smaller set than the polynomials used to determine \( {\mathcal{P}}_{n}^{0} \) and hence \( {\mathcal{... | Yes |
Lemma 68.2.5 If \( {s}_{k} \rightarrow h \), then for \( n \in \mathbb{N} \), there is a subsequence, still called \( {s}_{k} \) for which \( W{\left( {s}_{k}\right) }^{n} \rightarrow W{\left( h\right) }^{n} \) in \( {L}^{2}\left( \Omega \right) \) . | Proof: If \( {s}_{k} \rightarrow h \), does \( W{\left( {s}_{k}\right) }^{n} \rightarrow W{\left( h\right) }^{n} \) in \( {L}^{2}\left( \Omega \right) \) for some subsequence? First of all,\n\n\[ \n{\begin{Vmatrix}W\left( h\right) - W\left( {s}_{k}\right) \end{Vmatrix}}_{{L}^{2}\left( \Omega \right) }^{2} = {\left| {s}... | Yes |
Lemma 68.3.4 The following holds for \( f \in {\mathcal{E}}_{m} \)\n\n\[ \parallel \widetilde{f}{\parallel }_{{L}^{2}\left( {T}^{m}\right) } \leq \parallel f{\parallel }_{{L}^{2}\left( {T}^{m}\right) } \]\n\nalso\n\n\[ {I}_{m}\left( f\right) = {I}_{m}\left( \widetilde{f}\right) \] | Proof: This follows because, thanks to the properties of Lebesgue measure,\n\n\[ {\int }_{{T}^{m}}{\left| f\left( {t}_{1},\cdots ,{t}_{m}\right) \right| }^{2}d{t}_{1}\cdots d{t}_{m} = {\int }_{{T}^{m}}{\left| f\left( {t}_{{\sigma }_{1}},\cdots ,{t}_{{\sigma }_{m}}\right) \right| }^{2}d{t}_{1}\cdots d{t}_{m} \]\n\n\[ \e... | Yes |
Lemma 68.3.5 Let \( \left\{ {{A}_{1},\cdots ,{A}_{m}}\right\} \) be pairwise disjoint sets in \( \mathcal{B}\left( T\right) \) each having finite measure. Then the products \( {A}_{{i}_{1}} \times \cdots \times {A}_{{i}_{n}} \) are pairwise disjoint. Also to say that the function \[ \left( {{t}_{1},\cdots ,{t}_{n}}\rig... | Proof: Suppose the condition that the \( {A}_{k} \) are pairwise disjoint holds and consider two of these products, \( {A}_{{i}_{1}} \times \cdots \times {A}_{{i}_{n}} \) and \( {A}_{{j}_{1}} \times \cdots \times {A}_{{j}_{n}} \) . If the two ordered lists \( \left( {{i}_{1},\cdots ,{i}_{n}}\right) ,\left( {{j}_{1},\cd... | Yes |
Lemma 68.3.9 \( {I}_{n} \) is linear on \( {\mathcal{E}}_{n} \) . If \( f \in {\mathcal{E}}_{n} \) and \( \sigma \) is a permutation of \( \left( {1,\cdots, n}\right) \) and\n\n\[ \n{f}_{\sigma }\left( {{t}_{1},\cdots ,{t}_{n}}\right) \equiv f\left( {{t}_{\sigma \left( 1\right) },\cdots ,{t}_{\sigma \left( {t}_{n}\righ... | Proof: It is clear from the definition being well defined that \( {I}_{n} \) is linear. In particular, consider\n\n\[ \n{I}_{n}\left( {a\mathop{\sum }\limits_{\mathbf{i}}{c}_{\mathbf{i}}{\mathcal{X}}_{{A}_{{i}_{1}} \times \cdots \times {A}_{{i}_{n}}} + b\mathop{\sum }\limits_{\mathbf{j}}{d}_{\mathbf{j}}{\mathcal{X}}_{{... | Yes |
Lemma 68.3.10 Let \( {\mathcal{B}}_{0}\left( T\right) \) be the Borel sets having finite measure. Linear combinations of functions of the form\n\n\[ \n{\mathcal{X}}_{{A}_{1} \times \cdots \times {A}_{n}} \n\]\n\nwhere \( {A}_{i} \in {\mathcal{B}}_{0}\left( T\right) \) are dense in \( {L}^{2}\left( {T,{\mathcal{B}}^{n}}... | Proof: If you have \( U = {A}_{1} \times \cdots \times {A}_{n} \) in \( {T}^{n} \) one can approximate \( {\mathcal{X}}_{U \cap {R}_{p}} \) for \( {R}_{p} \equiv {\left( -p, p\right) }^{n} \) in \( {L}^{2} \) with linear combinations of sets of the desired form. In fact, you just consider \( {\mathcal{X}}_{{A}_{1} \cap... | Yes |
Lemma 68.3.11 The functions in \( {\mathcal{E}}_{n} \) mentioned above are dense in \( {L}^{2}\left( {T}^{n}\right) \) . | Proof: From Lemma 68.3.10, it suffices to show that \( {\mathcal{X}}_{{A}_{1} \times \cdots \times {A}_{n}} \) can be approximated in \( {L}^{2}\left( {T}^{n}\right) \) with functions in \( {\mathcal{E}}_{n} \) . This is where it will be important that the measure is sufficiently like Lebesgue measure. Let \( {\left\{ ... | Yes |
Theorem 68.3.12 The integral \( {I}_{n} \) defined on \( {\mathcal{E}}_{n} \) extends uniquely to an integral \( {I}_{n} \) defined on \( {L}^{2}\left( {T}^{n}\right) \) . This integral satisfies\n\n\[ \n{I}_{n}\left( f\right) \in {L}^{2}\left( \Omega \right) \n\]\n\nAlso\n\[ \nE\left( {{I}_{n}\left( f\right) {I}_{n}\l... | Proof: This follows right away from the density of \( {\mathcal{E}}_{n} \) in \( {L}^{2}\left( {T}^{n}\right) \) and the inequality 68.3.20. | No |
Lemma 68.3.14 Let \( n \) be given. Then \( { \cup }_{p \leq n}\left\{ {{I}_{p}\left( f\right) : f \in {\mathcal{E}}_{p}}\right\} \) is dense in \( {\mathcal{P}}_{n} = \) \( { \oplus }_{i = 0}^{n}{\mathcal{H}}_{i} \) . Consequently, every \( f \in {L}^{2}\left( {\Omega ,\mathcal{F}}\right) \) may be written as an infin... | Proof: It only remains to verify that \( {g}_{k} \) can be symmetric. However, this is obvious because if \( {g}_{k} \) is replaced with \( {\widetilde{g}}_{k} \) the integral \( {I}_{k} \) is unchanged. | No |
Lemma 68.4.3 Let \( \mathcal{P} \) denote the set of all polynomials in \( W\left( h\right) \) for \( h \in H \) . Then \( \mathcal{P} \) is dense in \( {L}^{p}\left( \Omega \right) \) . | Proof: Let \( g \in {L}^{{p}^{\prime }}\left( \Omega \right) \) and suppose that for every \( f \in D,{\int }_{\Omega }{gfdP} = 0 \) . Does it follow that \( g = 0 \) ? If so, then by the Riesz representation theorem, \( \mathcal{P} \) is dense in \( {L}^{p}\left( \Omega \right) \) . From Lemma 64.6.4, for a given \( h... | Yes |
Lemma 68.4.5 Functions of the form \( \mathop{\sum }\limits_{{k = 1}}^{n}{F}_{k}{h}_{k} \) where \( {F}_{k} \) is a polynomial in the \( W\left( h\right) \left( {{F}_{j} \in \mathcal{P}}\right) \) are dense in \( {L}^{p}\left( {\Omega, H}\right) \) for any \( p > 1 \) . Also each function of this form is in \( {D\delta... | \[ \delta \left( {\mathop{\sum }\limits_{{j = 1}}^{m}{F}_{j}{h}_{j}}\right) = \mathop{\sum }\limits_{{j = 1}}^{m}\delta \left( {{F}_{j}{h}_{j}}\right) = \mathop{\sum }\limits_{{j = 1}}^{m}{F}_{j}W\left( {h}_{j}\right) - \left\langle {D{F}_{j},{h}_{j}}\right\rangle \] | No |
Lemma 68.4.7 Let \( \Phi : \left\lbrack {0, T}\right\rbrack \times \Omega \rightarrow E \), be \( \mathcal{B}\left( \left\lbrack {0, T}\right\rbrack \right) \times \mathcal{F} \) measurable and suppose\n\n\[ \Phi \in K \equiv {L}^{p}\left( {\left\lbrack {0, T}\right\rbrack \times \Omega ;E}\right), p \geq 1 \]\n\nThen ... | Also, each \( \Phi \left( {t}_{j}^{k}\right) ,\Phi \left( {t}_{j - 1}^{k}\right) \) is in \( {L}^{p}\left( {\Omega ;E}\right) \). One can also assume that \( \Phi \left( 0\right) = 0 \). The mesh points \( {\left\{ {t}_{j}^{k}\right\} }_{j = 0}^{{m}_{k}} \) can be chosen to miss a given set of measure zero. In addition... | Yes |
Proposition 69.0.1 Suppose \( V \) is reflexive and a subset of \( H \) a separable Hilbert space with the inclusion map continuous. Suppose also that \( V \) is dense in \( H \) . Then identifying \( H \) and \( {H}^{\prime } \), it follows that \( H \) is dense in \( {V}^{\prime } \) and \( V \) is separable. | Proof: If \( H \) is not dense in \( {V}^{\prime } \), then by the Hahn Banach theorem, there exists \( {\phi }^{* * } \in {V}^{\prime \prime } \) such that \( {\phi }^{* * }\left( H\right) = 0 \) but \( {\phi }^{* * }\left( {\phi }^{ * }\right) \neq 0 \) for some \( {\phi }^{ * } \in {V}^{\prime } \smallsetminus \bar{... | Yes |
Proposition 69.0.3 Denote by \( \mathcal{B}\left( X\right) \) the Borel sets of \( X \) where \( X \) is any separable Banach space. Then\n\n\[ \mathcal{B}\left( X\right) = \sigma \left( {X}^{\prime }\right) \]\n\nHere \( \sigma \left( {X}^{\prime }\right) \) is the smallest \( \sigma \) algebra such that each \( \phi ... | Proof: By Lemma 21.1.6 there exists a countable subset of the unit ball in \( {X}^{\prime } \)\n\n\[ {\left\{ {\phi }_{n}\right\} }_{n = 1}^{\infty } = {D}^{\prime } \]\n\nsuch that\n\n\[ \parallel v{\parallel }_{X} = \sup \left\{ {\left| {\phi \left( v\right) }\right| : \phi \in {D}^{\prime }}\right\} .\n\]\n\nConside... | Yes |
Proposition 69.0.4 Let \( X \subseteq Y, X \) dense in \( Y \) and suppose \( X, Y \) are Banach spaces and that \( X \) is reflexive. Then \( X \in \mathcal{B}\left( Y\right) \) . | Proof: Define the functional\n\n\[ \phi \left( x\right) \equiv \left\{ \begin{array}{l} \parallel x{\parallel }_{X}\text{ if }x \in X \\ \infty \text{ if }x \in Y \smallsetminus X \end{array}\right. \]\n\nThen \( \phi \) is lower semicontinuous on \( Y \) . Here is why. Suppose \( \left( {x, a}\right) \notin \operatorn... | Yes |
Lemma 69.1.1 It is possible to consider \( {L}^{p}\left( D\right) \equiv V \) as a dense subspace of \( {\left( {H}_{0}^{1}\right) }^{\prime } \equiv H \) as follows. For \( f \in {L}^{p}\left( D\right) \) and \( \phi \in {H}_{0}^{1}\left( D\right) \) , \[ \langle f,\phi \rangle \equiv {\int }_{D}f\left( x\right) \phi ... | Proof: First of all, note that by 69.1.1 \[ \left| {\langle f,\phi \rangle }\right| \leq \parallel f{\parallel }_{{L}^{p}}\parallel \phi {\parallel }_{{L}^{{p}^{\prime }}} \leq C\parallel f{\parallel }_{{L}^{p}}\parallel \phi {\parallel }_{{H}_{0}^{1}} \] and so it is certainly possible to consider \( {L}^{p} \subseteq... | Yes |
Lemma 69.2.4 Let \( \bar{f} \) be as defined in Definition 69.2.2. Then for \( f \in {L}^{p}\left( {a, b;X}\right) \) for \( p \in \lbrack 1,\infty ) \) , \[ \mathop{\lim }\limits_{{\delta \rightarrow 0}}{\int }_{a}^{b}\parallel \bar{f}\left( {t - \delta }\right) - f\left( t\right) {\parallel }_{X}^{p}{dt} = 0. \] | Proof: Regarding the measure space as \( \left( {a, b}\right) \) with Lebesgue measure, by regularity of the measure, there exists \( g \in {C}_{c}\left( {a, b;X}\right) \) such that \( \parallel f - g{\parallel }_{p} < \varepsilon \) . Here the norm is the norm in \( {L}^{p}\left( {a, b;X}\right) \) . Therefore, \[ {\... | Yes |
Lemma 69.2.6 The above definition is well defined. | Proof: Suppose both \( h \) and \( g \) work in the definition. Then for all \( \phi \in {C}_{c}^{\infty }\left( {a, b}\right) \) ,\n\n\[ \n{\int }_{a}^{b}\left( {h\left( t\right) - g\left( t\right) }\right) \phi \left( t\right) {dt} = 0.\n\]\n\nTherefore, by Lemma 69.2.1, \( h\left( t\right) - g\left( t\right) = 0 \) ... | Yes |
Lemma 69.2.8 Suppose \( f \in {L}^{1}\left( {a, b;X}\right) \) and for all \( \phi \in {C}_{c}^{\infty }\left( {a, b}\right) \) ,\n\n\[{\int }_{a}^{b}f\left( t\right) {\phi }^{\prime }\left( t\right) {dt} = 0\]\n\nThen there exists a constant, \( a \in X \) such that \( f\left( t\right) = a \) a.e. | Proof: Let \( {\phi }_{0} \in {C}_{c}^{\infty }\left( {a, b}\right) ,{\int }_{a}^{b}{\phi }_{0}\left( x\right) {dx} = 1 \) and define for \( \phi \in {C}_{c}^{\infty }\left( {a, b}\right) \)\n\n\[{\psi }_{\phi }\left( x\right) \equiv {\int }_{a}^{x}\left\lbrack {\phi \left( t\right) - \left( {{\int }_{a}^{b}\phi \left(... | Yes |
Theorem 69.2.9 Suppose \( f,{f}^{\prime } \) both are in \( {L}^{1}\left( {a, b;X}\right) \) where the derivative is taken in the sense of \( X \) valued distributions. Then there exists a unique point of \( X \) , denoted by \( f\left( a\right) \) such that the following formula holds a.e. \( t \) .\n\n\[ f\left( t\ri... | Proof:\n\n\[ {\int }_{a}^{b}\left( {f\left( t\right) - {\int }_{a}^{t}{f}^{\prime }\left( s\right) {ds}}\right) {\phi }^{\prime }\left( t\right) {dt} = {\int }_{a}^{b}f\left( t\right) {\phi }^{\prime }\left( t\right) {dt} - {\int }_{a}^{b}{\int }_{a}^{t}{f}^{\prime }\left( s\right) {\phi }^{\prime }\left( t\right) {dsd... | Yes |
Corollary 69.2.10 Suppose \( f,{f}^{\prime } \in {L}^{1}\left( {a, b;X}\right) \) and suppose \( \phi \in {C}^{1}\left( \left\lbrack {a, b}\right\rbrack \right) \) . Then the following integration by parts formula holds. | \[ {\int }_{a}^{b}f\left( t\right) {\phi }^{\prime }\left( t\right) {dt} = f\left( b\right) \phi \left( b\right) - f\left( a\right) \phi \left( a\right) - {\int }_{a}^{b}{f}^{\prime }\left( t\right) \phi \left( t\right) {dt}. \] Proof: From Theorem 69.2.9 \[ {\int }_{a}^{b}f\left( t\right) {\phi }^{\prime }\left( t\rig... | Yes |
Lemma 69.2.11 Let \( \bar{f} \) be given in Definition 69.2.2 and suppose \( f,{f}^{\prime } \in {L}^{1}\left( {a, b;X}\right) \) . Then \( \bar{f},{\bar{f}}^{\prime } \in {L}^{1}\left( {{2a} - b,{2b} - a;X}\right) \) also and \[ {\bar{f}}^{\prime }\left( t\right) \equiv \left\{ \begin{array}{l} {f}^{\prime }\left( t\r... | Proof: It is clear from the definition of \( \bar{f} \) that \( \bar{f} \in {L}^{1}\left( {{2a} - b,{2b} - a;X}\right) \) and that in fact \[ \parallel \bar{f}{\parallel }_{{L}^{1}\left( {{2a} - b,{2b} - a;X}\right) } \leq 3\parallel f{\parallel }_{{L}^{1}\left( {a, b;X}\right) }. \] \( \left( {69.2.9}\right) \) Let \(... | Yes |
Theorem 69.2.13 Let \( V \) and \( H \) be a Banach space and Hilbert space as described in Definition 69.2.12. Suppose \( f \in {L}^{p}\left( {0, T;V}\right) \) and \( {f}^{\prime } \in {L}^{{p}^{\prime }}\left( {0, T;{V}^{\prime }}\right) \) . Then \( f \) is a.e. equal to a continuous function mapping \( \left\lbrac... | Proof: Let \( \Psi \in {C}_{c}^{\infty }\left( {-T,{2T}}\right) \) satisfy \( \Psi \left( t\right) = 1 \) if \( t \in \left\lbrack {-T/2,{3T}/2}\right\rbrack \) and \( \Psi \left( t\right) \geq 0 \) . For \( t \in \mathbb{R} \), define \[ \widehat{f}\left( t\right) \equiv \left\{ \begin{array}{l} \bar{f}\left( t\right)... | Yes |
Theorem 69.3.2 Let \( V \subseteq H = {H}^{\prime } \subseteq {V}^{\prime } \) be a Gelfand triple and suppose \( Y \in \) \( {L}^{{p}^{\prime }}\left( {0, T;{V}^{\prime }}\right) \equiv {K}^{\prime } \) and\n\n\[ \nX\left( t\right) = {X}_{0} + {\int }_{0}^{t}Y\left( s\right) {ds}\text{ in }{V}^{\prime } \n\]\n\n(69.3.... | Proof: By Lemma 65.3.1, there exists a sequence of uniform partitions \( {\left\{ {t}_{k}^{n}\right\} }_{k = 0}^{{m}_{n}} = \) \( {\mathcal{P}}_{n},{\mathcal{P}}_{n} \subseteq {\mathcal{P}}_{n + 1} \), of \( \left\lbrack {0, T}\right\rbrack \) such that the step functions\n\n\[ \n\mathop{\sum }\limits_{{k = 0}}^{{{m}_{... | Yes |
Lemma 69.3.3 Let \( s < t \) . Then for \( X, Y \) satisfying 69.3.15\n\n\[{\left| X\left( t\right) \right| }^{2} = {\left| X\left( s\right) \right| }^{2} + 2{\int }_{s}^{t}\langle Y\left( u\right), X\left( t\right) \rangle {du} - {\left| X\left( t\right) - X\left( s\right) \right| }^{2}\] | Proof: It follows from the following computations\n\n\[X\left( t\right) - X\left( s\right) = {\int }_{s}^{t}Y\left( u\right) {du}\]\n\n\[- {\left| X\left( t\right) - X\left( s\right) \right| }^{2} = - {\left| X\left( t\right) \right| }^{2} + 2\left( {X\left( t\right), X\left( s\right) }\right) - {\left| X\left( s\right... | Yes |
Lemma 69.3.4 In the above situation, \[ \mathop{\sup }\limits_{{t \in \left\lbrack {0, T}\right\rbrack }}{\left| X\left( t\right) \right| }_{H} \leq C\left( {\parallel Y{\parallel }_{{K}^{\prime }},\parallel X{\parallel }_{K}}\right) \] Also, \( t \rightarrow X\left( t\right) \) is weakly continuous with values in \( H... | Proof: From the above formula applied to the \( {k}^{\text{th }} \) partition of \( \left\lbrack {0, T}\right\rbrack \) described above, \[ {\left| X\left( {t}_{m}\right) \right| }^{2} - {\left| {X}_{0}\right| }^{2} = \mathop{\sum }\limits_{{j = 0}}^{{m - 1}}{\left| X\left( {t}_{j + 1}\right) \right| }^{2} - {\left| X\... | Yes |
Lemma 69.4.1 Suppose \( V, W \) are separable Banach spaces, \( W \) also a Hilbert space such that \( V \) is dense in \( W \) and \( B \in \mathcal{L}\left( {W,{W}^{\prime }}\right) \) satisfies\n\n\[ \langle {Bx}, x\rangle \geq 0,\langle {Bx}, y\rangle = \langle {By}, x\rangle, B \neq 0. \]\n\nThen there exists a co... | Proof: Let \( {\left\{ {g}_{k}\right\} }_{k = 1}^{\infty } \) be linearly independent vectors of \( V \) whose span is dense in \( V \) . This is possible because \( V \) is separable. Thus, their span is also dense in \( W \) . Let \( {n}_{1} \) be the first index such that \( \left\langle {B{g}_{{n}_{1}},{g}_{{n}_{1}... | Yes |
Theorem 69.4.2 Let \( V \subseteq W,{W}^{\prime } \subseteq {V}^{\prime } \) be separable Banach spaces, \( W \) a separable Hilbert space, and let \( Y \in {L}^{{p}^{\prime }}\left( {0, T;{V}^{\prime }}\right) \equiv {K}^{\prime } \) and\n\n\[ \n{BX}\left( t\right) = B{X}_{0} + {\int }_{0}^{t}Y\left( s\right) {ds}\tex... | Proof: By Lemma 65.3.1, there exists a sequence of uniform partitions \( {\left\{ {t}_{k}^{n}\right\} }_{k = 0}^{{m}_{n}} = \) \( {\mathcal{P}}_{n},{\mathcal{P}}_{n} \subseteq {\mathcal{P}}_{n + 1} \), of \( \left\lbrack {0, T}\right\rbrack \) such that the step functions\n\n\[ \n\mathop{\sum }\limits_{{k = 0}}^{{{m}_{... | Yes |
Lemma 69.4.3 Let \( s < t \) . Then for \( X, Y \) satisfying 69.4.23\n\n\[ \langle {BX}\left( t\right), X\left( t\right) \rangle = \langle {BX}\left( s\right), X\left( s\right) \rangle \]\n\n\[ + 2{\int }_{s}^{t}\langle Y\left( u\right), X\left( t\right) \rangle {du} - \langle B\left( {X\left( t\right) - X\left( s\rig... | Proof: It follows from the following computations\n\n\[ B\left( {X\left( t\right) - X\left( s\right) }\right) = {\int }_{s}^{t}Y\left( u\right) {du} \]\n\nand so\n\n\[ 2{\int }_{s}^{t}\langle Y\left( u\right), X\left( t\right) \rangle {du} - \langle B\left( {X\left( t\right) - X\left( s\right) }\right) ,\left( {X\left(... | Yes |
Theorem 69.5.2 Let \( E \subseteq W \subseteq X \) where the injection map is continuous from \( W \) to \( X \) and compact from \( E \) to \( W \) . Then for every \( \varepsilon > 0 \) there exists a constant, \( {C}_{\varepsilon } \) such that for all \( u \in E \) , \[ \parallel u{\parallel }_{W} \leq \varepsilon ... | Proof: Suppose not. Then there exists \( \varepsilon > 0 \) and for each \( n \in \mathbb{N},{u}_{n} \) such that \[ {\begin{Vmatrix}{u}_{n}\end{Vmatrix}}_{W} > \varepsilon {\begin{Vmatrix}{u}_{n}\end{Vmatrix}}_{E} + n{\begin{Vmatrix}{u}_{n}\end{Vmatrix}}_{X} \] Now let \( {v}_{n} = {u}_{n}/{\begin{Vmatrix}{u}_{n}\end{... | Yes |
Corollary 69.5.5 Let \( E \subseteq W \subseteq X \) where the injection map is continuous from \( W \) to \( X \) and compact from \( E \) to \( W \) . Then if \( \gamma > \alpha \), the embedding of \( {C}^{0,\gamma }\left( {\left\lbrack {0, T}\right\rbrack, E}\right) \) into \( {C}^{0,\alpha }\left( {\left\lbrack {0... | Proof: Let \( \phi \in {C}^{0,\gamma }\left( {\left\lbrack {0, T}\right\rbrack, E}\right) \)\n\n\[ \frac{\parallel \phi \left( t\right) - \phi \left( s\right) {\parallel }_{X}}{{\left| t - s\right| }^{\alpha }} \leq {\left( \frac{\parallel \phi \left( t\right) - \phi \left( s\right) {\parallel }_{W}}{{\left| t - s\righ... | Yes |
Lemma 70.2.2 Let \( \left\{ {\mathbf{f}}_{n}\right\} \) be a sequence in \( X \) and suppose that the \( {k}^{\text{th }} \) components \( {f}_{nk} \) are bounded in \( {C}^{0,1}\left( \left\lbrack {0, T}\right\rbrack \right) \) . (This refers to the Hölder space with \( \gamma = 1 \) .) Then there exists a subsequence... | Proof: By the Ascoli-Arzelà theorem, there exists a subsequence \( {n}_{1} \) such that the first component \( {f}_{{n}_{1}1} \) converges in \( C\left( \left\lbrack {0, T}\right\rbrack \right) \) . Then taking a subsequence, one can obtain \( {n}_{2} \) a subsequence of \( {n}_{1} \) such that both the first and secon... | Yes |
Lemma 70.2.6 Suppose, \( {u}_{n\left( \omega \right) } \rightarrow u \) weakly in \( {L}^{{p}^{\prime }}\left( {\left\lbrack {0, T}\right\rbrack ;{V}^{\prime }}\right) \) where \( u \) is product measurable measurable and \( \left\{ {u}_{n\left( \omega \right) }\right\} \) is a subsequence of \( \left\{ {u}_{n}\right\}... | Proof: For \( f, g \in {L}^{{p}^{\prime }}\left( {\left\lbrack {0, T}\right\rbrack ;{V}^{\prime }}\right) \equiv {\mathcal{V}}^{\prime },{L}^{p}\left( {\left\lbrack {0, T}\right\rbrack ;V}\right) \equiv \mathcal{V} \), let \( \left\{ {\phi }_{k}\right\} \) be a countable dense subset of \( {L}^{p}\left( {\left\lbrack {... | Yes |
Corollary 70.2.7 Let \( V \) be a reflexive separable Banach space and \( {V}^{\prime } \) its dual and \( \frac{1}{p} + \frac{1}{{p}^{\prime }} = 1 \) where \( p > 1 \) as usual. Let the functions \( t \rightarrow {u}_{n\left( \omega \right) }\left( {t,\omega }\right) \) be in \( {L}^{{p}^{\prime }}\left( {\left\lbrac... | Proof: It suffices to consider the functions \( {v}_{n}\left( {t,\omega }\right) \equiv {u}_{n\left( \omega \right) }\left( {t,\omega }\right) \) and use the result of Theorem 70.2.1. | No |
Lemma 70.3.1 Suppose \( \mathbf{N}\left( {t,\mathbf{u},\mathbf{v},\mathbf{w},\omega }\right) \in {\mathbb{R}}^{d} \) for \( \mathbf{u},\mathbf{v},\mathbf{w} \in {\mathbb{R}}^{d}, t \in \left\lbrack {0, T}\right\rbrack \) and \( \left( {t,\mathbf{u},\mathbf{v},\mathbf{w},\omega }\right) \rightarrow \mathbf{N}\left( {t,\... | Proof: Let \( {\mathbf{u}}_{n} \) be the solution to the following equation:\n\n\[ {\mathbf{u}}_{n}\left( {t,\omega }\right) - {\mathbf{u}}_{0}\left( \omega \right) + {\int }_{0}^{t}\mathbf{N}\left( {s,{\tau }_{1/n}{\mathbf{u}}_{n}\left( {s,\omega }\right) ,{\mathbf{u}}_{n}\left( {s - h,\omega }\right) ,{\tau }_{1/n}{\... | Yes |
Theorem 70.3.3 Suppose \( \mathbf{N}\left( {t,\mathbf{u},\mathbf{v},\mathbf{w},\omega }\right) \in {\mathbb{R}}^{d} \) for \( \mathbf{u},\mathbf{v},\mathbf{w} \in {\mathbb{R}}^{d}, t \in \left\lbrack {0, T}\right\rbrack \) and \( \left( {t,\mathbf{u},\mathbf{v},\mathbf{w},\omega }\right) \rightarrow \mathbf{N}\left( {t... | Proof: Let \( {P}_{m} \) denote the projection onto the closed ball \( \overline{B\left( {\mathbf{0},{9}^{m}}\right) } \) . Then from the above lemma, there exists a product measurable solution \( {\mathbf{u}}_{m} \) to the integral equation\n\n\[ {\mathbf{u}}_{m}\left( {t,\omega }\right) - {\mathbf{u}}_{0}\left( \omeg... | No |
Lemma 70.4.3 For fixed \( \omega \in \Omega ,\widehat{\mathbf{f}} \in {L}^{2}\left( {\left\lbrack {0, T}\right\rbrack ;{W}^{\prime }}\right) \), and\n\n\[ \left( {t,\mathbf{w}}\right) \rightarrow B\left( {\mathbf{w},\mathbf{q}\left( {t,\omega }\right) }\right) ,\left( {t,\mathbf{w}}\right) \rightarrow B\left( {\mathbf{... | Proof: The first claim is straightforward to prove from the definition of \( A \) and \( N \) . Consider the next claim about continuity. Let \( \mathbf{z} \in W \) be given. Then from the fact that all the functions are divergence free,\n\n\[ \left| {\langle B\left( {\mathbf{w},\mathbf{q}\left( t\right) }\right) - B\l... | Yes |
Lemma 70.4.4 Let \( {\mathbf{u}}_{0} \) have values in \( H \) and be \( \mathcal{F} \) measurable, and let \( {\mathbf{u}}_{n} \) be a solution to 70.4.6. Then for each \( \omega \), the estimate 70.4.8 holds. Also there is a subsequence, still called \( {\mathbf{u}}_{n} \) such that the convergence for 70.4.9 - 70.4.... | Proof: All that remains to show is the last claim about weak continuity into \( H \) . The equation 70.4.13 shows that \( \mathbf{u}\left( {\cdot ,\omega }\right) \) is continuous into \( {V}^{\prime } \) . However, the weak convergence and the estimate 70.4.8 show that \( \mathbf{u}\left( {\cdot ,\omega }\right) \) is... | Yes |
Theorem 70.4.5 Let \( \mathbf{f}\left( {t,\omega }\right) ,\mathbf{q}\left( {t,\omega }\right) \) be product measurable and \( {\mathbf{u}}_{0} \) be measurable, such that for each \( \omega \in \Omega ,\mathbf{f}\left( {\cdot ,\omega }\right) \in {L}^{2}\left( {\left\lbrack {0, T}\right\rbrack ;{W}^{\prime }}\right) ,... | Proof: Letting \( {\mathbf{u}}_{n} \) be a solution to 70.4.6, we verify the conditions of Theorem 70.2 .8 for \( {\mathbf{u}}_{n} \) .\n\nThe assumption in this theorem that the \( {\mathbf{u}}_{n} \) are bounded follows from the above estimate 70.4.8. Then it was shown in the above lemma that whenever a sequence sati... | Yes |
Theorem 70.4.6 Let \( \mathbf{f}\left( {t,\omega }\right) ,\mathbf{q}\left( {t,\omega }\right) \) be product measurable and \( {\mathbf{u}}_{0} \) be measurable, such that for each \( \omega \in \Omega ,\mathbf{f}\left( {\cdot ,\omega }\right) \in {L}^{2}\left( {\left\lbrack {0, T}\right\rbrack ;{W}^{\prime }}\right) ,... | Proof: The last claim follows from the estimates used in the Galerkin method, taking expectations and passing to a limit. To verify the continuity into \( H \), one can observe that from the integral equation, \( \mathbf{u} \) is continuous into \( {V}^{\prime } \) . One has\n\n\[ {\left| \mathbf{u}\left( t\right) \rig... | Yes |
Theorem 70.5.3 For each \( \omega \) let \( {\mathbf{u}}_{0}\left( \omega \right) \in V,{\mathbf{v}}_{0}\left( \omega \right) \in H \) . Let \( \mathbf{f} \in {\mathcal{V}}^{\prime } \) . Also assume the gap \( g \) and sliding velocity \( {\dot{\mathbf{U}}}_{T} \) are \( \mathcal{F} \) measurable. Then there exists a ... | It only remains to check the last claim about measurability into the other spaces. By density of \( V \) into \( H \), it follows that \( {H}^{\prime } \) is dense in \( {V}^{\prime } \) and so a simple Pettis theorem argument implies right away that \( \omega \rightarrow \mathbf{v}\left( {t,\omega }\right) \) is \( \m... | No |
Theorem 71.1.1 Suppose \( \mathbf{N}\left( {t,\mathbf{u},\mathbf{v},\mathbf{w},\omega }\right) \in {\mathbb{R}}^{d} \) for \( \mathbf{u},\mathbf{v},\mathbf{w} \in {\mathbb{R}}^{d}, t \in \left\lbrack {0, T}\right\rbrack \) and \( \left( {t,\mathbf{u},\mathbf{v},\mathbf{w},\omega }\right) \rightarrow \mathbf{N}\left( {t... | Proof: Let \( 0 = {t}_{0} < {t}_{1} < \cdots < {t}_{n} = T \) . From Theorem 70.3.3, there exists a solution to the integral equation \( \mathbf{u} \) which has the property that \( \mathbf{u}\left( {t \land {t}_{j}}\right) \) is \( {\mathcal{F}}_{{t}_{j}} \) measurable. One simply applies this theorem to the successio... | Yes |
Theorem 71.2.1 Suppose \( \mathbf{N}\left( {t,\mathbf{u},\mathbf{v},\mathbf{w},\omega }\right) \in {\mathbb{R}}^{d} \) for \( \mathbf{u},\mathbf{v},\mathbf{w} \in {\mathbb{R}}^{d}, t \in \left\lbrack {0, T}\right\rbrack \) and \( \left( {t,\mathbf{u},\mathbf{v},\mathbf{w},\omega }\right) \rightarrow \mathbf{N}\left( {t... | Proof: The only thing left is to observe that the given estimate is sufficient to obtain an estimate for the solutions to the integral equation for \( \widehat{\mathbf{u}} \) defined above. Then from Theorem 71.1.1, there exists a unique progressively measurable solution for \( \widehat{\mathbf{u}} \) and hence for \( ... | No |
Theorem 71.2.2 Suppose the weak monotonicity condition 71.2.5 and the growth estimate 71.2.3. Also assume \( \mathbf{N}\left( {t,\mathbf{u},\mathbf{v},\mathbf{w},\omega }\right) \in {\mathbb{R}}^{d} \) for \( \mathbf{u},\mathbf{v},\mathbf{w} \in {\mathbb{R}}^{d}, t \in \left\lbrack {0, T}\right\rbrack \) and \( \left( ... | Proof: It only remains to verify the uniqueness assertion. This happens because the fixed point is unique in \( {L}^{2}\left( {\Omega, C\left( {\left\lbrack {0, T}\right\rbrack ,{\mathbb{R}}^{n}}\right) }\right) \) . Therefore, off a set of measure zero the two solutions are equal for all \( t \) . | Yes |
Theorem 71.3.1 Suppose 71.3.11, 71.3.13, 71.3.15, 71.3.12 and let\n\n\[ w\left( t\right) = {w}_{0} + {\int }_{0}^{t}u\left( s\right) {ds},{w}_{0} \in {L}^{2}\left( \Omega \right) ,{w}_{0}\text{is}{\mathcal{F}}_{0}\text{measurable.} \]\n\nThen there exists a unique progressively measurable solution \( u \) to the integr... | Proof: Let \( v \in {L}^{2}\left( {\Omega ;C\left( {\left\lbrack {0, T}\right\rbrack ;H}\right) }\right) \) where \( v \) is also progressively measurable. Then let \( u \) be given by\n\n\[ u\left( {t,\omega }\right) - {u}_{0}\left( \omega }\right) + {\int }_{0}^{t}N\left( {s, v\left( {s,\omega }\right), v\left( {s - ... | Yes |
Theorem 71.3.2 Suppose 71.3.11, 71.3.14, 71.3.15,71.3.13, 71.3.19 and let\n\n\[ w\left( t\right) = {w}_{0} + {\int }_{0}^{t}u\left( s\right) {ds},{w}_{0} \in {L}^{2}\left( \Omega \right) ,{w}_{0}\text{is}{\mathcal{F}}_{0}\text{measurable.} \]\n\nThen there exists a unique progressively measurable solution \( u \) to th... | Proof: Let \( {u}_{n} \) be the unique solution to the integral equation\n\n\[ {u}_{n}\left( {t,\omega }\right) - {u}_{0}\left( \omega }\right) + {\int }_{0}^{t}N\left( {s,{P}_{n}{u}_{n}\left( {s,\omega }\right) ,{P}_{n}{u}_{n}\left( {s - h,\omega }\right) ,{P}_{n}{w}_{n}\left( {s,\omega }\right) ,\omega }\right) {ds} ... | Yes |
Proposition 72.1.2 Let \( X\left( t\right) \) be a stochastic process having values in \( E \) a complete metric space and let it be \( {\mathcal{F}}_{t} \) adapted and left continuous where \( {\mathcal{F}}_{t} \) is a normal filtration. Then it is predictable. If \( t \rightarrow X\left( {t,\omega }\right) \) is cont... | Proof: First suppose \( X \) is continuous for all \( \omega \in \Omega \) . Define\n\n\[ \n{I}_{m, k} \equiv \left( {\left( {k - 1}\right) {2}^{-m}T, k{2}^{-m}T}\right\rbrack \n\]\n\nif \( k \geq 1 \) and \( {I}_{m,0} = \{ 0\} \) if \( k = 1 \) . Then define\n\n\[ \n{X}_{m}\left( t\right) \equiv \mathop{\sum }\limits_... | Yes |
Lemma 72.2.2 Let \( {f}_{n} \rightarrow f \) in \( {L}^{p}\left( {\left\lbrack {0, T}\right\rbrack \times \Omega, E}\right) \). Then there exists a subsequence \( {n}_{k} \) and a set of measure zero \( N \) such that if \( \omega \notin N \), then\n\n\[ \n{f}_{{n}_{k}}\left( {\cdot ,\omega }\right) \rightarrow f\left(... | Proof: We have\n\n\[ \nP\left( \left\lbrack {{\begin{Vmatrix}{f}_{n} - f\end{Vmatrix}}_{{L}^{p}\left( {\left\lbrack {0, T}\right\rbrack, E}\right) } > \lambda }\right\rbrack \right) \leq \frac{1}{\lambda }{\int }_{\Omega }{\begin{Vmatrix}{f}_{n} - f\end{Vmatrix}}_{{L}^{p}\left( {\left\lbrack {0, T}\right\rbrack, E}\rig... | Yes |
Lemma 72.3.3 In Situation 72.3.1 the following formula holds for a.e. \( \omega \) for \( 0 < \) \( s < t \) where \( M\left( t\right) \equiv {\int }_{0}^{t}Z\left( u\right) {dW}\left( u\right) \) . Here and elsewhere, \( \left| \cdot \right| \) denotes the norm in \( H \) and \( \langle \cdot , \cdot \rangle \) denote... | Proof: The formula is a straight forward computation which holds a.e. \( \omega \) .\n\n\[ \n{\left| M\left( t\right) - M\left( s\right) \right| }^{2} - {\left| X\left( t\right) - X\left( s\right) - \left( M\left( t\right) - M\left( s\right) \right) \right| }^{2} + 2\left( {X\left( s\right), M\left( t\right) - M\left( ... | Yes |
Lemma 72.4.1 In the Situation 72.3.1, \[ E\left( {\mathop{\sup }\limits_{{t \in \left\lbrack {0, T}\right\rbrack }}{\left| X\left( t\right) \right| }_{H}^{2}}\right) < C\left( {\parallel Y{\parallel }_{{K}^{\prime }},\parallel X{\parallel }_{K},\parallel Z{\parallel }_{J},{\begin{Vmatrix}{X}_{0}\end{Vmatrix}}_{{L}^{2}\... | Proof: Consider the formula in Lemma 72.3.3. \[ {\left| X\left( t\right) \right| }^{2} = {\left| X\left( s\right) \right| }^{2} + 2{\int }_{s}^{t}\langle Y\left( u\right), X\left( t\right) \rangle {du} + 2\left( {X\left( s\right), M\left( t\right) - M\left( s\right) }\right) \[ + {\left| M\left( t\right) - M\left( s\ri... | Yes |
Lemma 72.5.1 Let \( X\left( s\right) - {X}_{k}^{l}\left( s\right) \equiv {\Delta }_{k}\left( s\right) \) . Then the following limit occurs.\n\n\[ \mathop{\lim }\limits_{{k \rightarrow \infty }}P\left( \left\lbrack {\mathop{\sup }\limits_{{t \in \left\lbrack {0, T}\right\rbrack }}\left| {{\int }_{0}^{t}\mathcal{R}\left(... | Proof: First note that from Lemma 72.4.1, for a.e. \( \omega, X\left( t\right) \) has values in \( H \) for \( t \in \left\lbrack {0, T}\right\rbrack \) and so it makes sense to consider it in the stochastic integral provided it is \( H \) progressively measurable. However, as noted in Situation 72.3.1, this function i... | Yes |
Theorem 72.6.2 In Situation 72.3.1, off a set of measure zero, for every \( t \in \left\lbrack {0, T}\right\rbrack \) , \[ {\left| X\left( t\right) \right| }^{2} = {\left| {X}_{0}\right| }^{2} + {\int }_{0}^{t}\left( {2\langle Y\left( s\right) ,\bar{X}\left( s\right) \rangle + \parallel Z\left( s\right) {\parallel }_{{... | Proof: Let \( t \notin D \) . For \( t > 0 \), let \( t\left( k\right) \) denote the largest point of \( {\mathcal{P}}_{k} \) which is less than \( t \) . Suppose \( t\left( m\right) < t\left( k\right) \) . Hence \( m \leq k \) . Then \[ X\left( {t\left( m\right) }\right) = {X}_{0} + {\int }_{0}^{t\left( m\right) }Y\le... | Yes |
Lemma 73.1.1 Let \( \Phi : \left\lbrack {0, T}\right\rbrack \times \Omega \rightarrow V \), be \( \mathcal{B}\left( \left\lbrack {0, T}\right\rbrack \right) \times \mathcal{F} \) measurable and suppose\n\n\[ \Phi \in K \equiv {L}^{p}\left( {\left\lbrack {0, T}\right\rbrack \times \Omega ;E}\right), p \geq 1 \]\n\nThen ... | Also, each \( \Phi \left( {t}_{j}^{k}\right) ,\Phi \left( {t}_{j - 1}^{k}\right) \) is in \( {L}^{p}\left( {\Omega ;E}\right) \). One can also assume that \( \Phi \left( 0\right) = 0 \). The mesh points \( {\left\{ {t}_{j}^{k}\right\} }_{j = 0}^{{m}_{k}} \) can be chosen to miss a given set of measure zero. In addition... | Yes |
Lemma 73.1.2 Let \( {f}_{n} \rightarrow f \) in \( {L}^{p}\left( {\left\lbrack {0, T}\right\rbrack \times \Omega, E}\right) \) . Then there exists a subsequence \( {n}_{k} \) and a set of measure zero \( N \) such that if \( \omega \notin N \), then\n\n\[ \n{f}_{{n}_{k}}\left( {\cdot ,\omega }\right) \rightarrow f\left... | Proof: We have\n\n\[ \nP\left( \left\lbrack {{\begin{Vmatrix}{f}_{n} - f\end{Vmatrix}}_{{L}^{p}\left( {\left\lbrack {0, T}\right\rbrack, E}\right) } > \lambda }\right\rbrack \right) \leq \frac{1}{\lambda }{\int }_{\Omega }{\begin{Vmatrix}{f}_{n} - f\end{Vmatrix}}_{{L}^{p}\left( {\left\lbrack {0, T}\right\rbrack, E}\rig... | Yes |
Theorem 73.3.2 Let \( Z \) be progressively measurable and in \[ {L}^{2}\left( {\left\lbrack {0, T}\right\rbrack \times \Omega ,{\mathcal{L}}_{2}\left( {{Q}^{1/2}U, W}\right) }\right) . \] Also suppose \( X \) is progressively measurable and in \( {L}^{2}\left( {\left\lbrack {0, T}\right\rbrack \times \Omega, W}\right)... | Proof: First suppose that \( \left\langle {{BX}\left( {t}_{k}^{n}\right), X\left( {t}_{k}^{n}\right) }\right\rangle \in {L}^{\infty }\left( \Omega \right) \) . Then \[ \left\langle {{BX}\left( {t}_{j}^{n}\right) ,{\int }_{{t}_{j}^{n} \land t}^{{t}_{j + 1}^{n} \land t}{ZdW}}\right\rangle \] is in \( {L}^{1}\left( \Omega... | Yes |
In Situation 73.2.1 the following formula holds for a.e. \( \omega \) for \( 0 < \) \( s < t \) where \( M\left( t\right) \equiv {\int }_{0}^{t}Z\left( u\right) {dW}\left( u\right) \) which has values in \( W \) . In the following, \( \langle \cdot , \cdot \rangle \) denotes the duality pairing between \( V,{V}^{\prime... | Proof: From the formula which is assumed to hold, \n\n\[ {BX}\left( t\right) = B{X}_{0} + {\int }_{0}^{t}Y\left( u\right) {du} + {BM}\left( t\right) \]\n\n\[ {BX}\left( s\right) = B{X}_{0} + {\int }_{0}^{s}Y\left( u\right) {du} + {BM}\left( s\right) \]\n\nThen \n\n\[ {BM}\left( t\right) - {BM}\left( s\right) + {\int }_... | Yes |
Lemma 73.4.2 In the Situation 73.2.1, the following holds. For a.e. t\n\n\[ E\left( {\langle {BX}\left( t\right), X\left( t\right) \rangle }\right) \]\n\n\[ < C\left( {\parallel Y{\parallel }_{{K}^{\prime }},\parallel X{\parallel }_{K},\parallel Z{\parallel }_{J},{\begin{Vmatrix}\left\langle B{X}_{0},{X}_{0}\right\rang... | Proof: Consider the formula in Lemma 73.4.1.\n\n\[ \langle {BX}\left( t\right), X\left( t\right) \rangle = \langle {BX}\left( s\right), X\left( s\right) \rangle \]\n\n\[ + 2{\int }_{s}^{t}\langle Y\left( u\right), X\left( t\right) \rangle {du} + \langle B\left( {M\left( t\right) - M\left( s\right) }\right), M\left( t\r... | Yes |
Lemma 73.6.2 There exists a subsequence still denoted with the subscript \( k \) and an enlarged set of measure zero \( N \) including the earlier one such that \( B{X}_{k}^{l}\left( t\right), B{X}_{k}^{r}\left( t\right) \) also converges pointwise a.e. \( t \) to \( {BX}\left( t\right) \) in \( {W}^{\prime } \) and \(... | Proof: To see that such a sequence exists, let \( {n}_{k} \) be such that\n\n\[ \n{\int }_{\Omega }{\int }_{0}^{T}{\begin{Vmatrix}B{X}_{{n}_{k}}^{r}\left( t\right) - BX\left( t\right) \end{Vmatrix}}_{{W}^{\prime }}^{2}{dtdP} + {\int }_{\Omega }{\int }_{0}^{T}{\begin{Vmatrix}{X}_{{n}_{k}}^{r}\left( t\right) - X\left( t\... | Yes |
Lemma 73.6.3 In the above context, let \( X\left( s\right) - {X}_{k}^{l}\left( s\right) \equiv {\Delta }_{k}\left( s\right) \). Then the integral\n\n\[ \n{\int }_{0}^{t}{\left( Z \circ {J}^{-1}\right) }^{ * }{BX} \circ {JdW} \]\n\nexists as a local martingale and the following limit is valid for the subsequence of Lemm... | Proof: In the argument \( {\tau }_{m} \) will be defined in 73.6.20. Let\n\n\[ \n{A}_{k} \equiv \left\{ {\omega : \mathop{\sup }\limits_{{t \in \left\lbrack {0, T}\right\rbrack }}\left| {{\int }_{0}^{t}{\left( Z \circ {J}^{-1}\right) }^{ * }B{\Delta }_{k} \circ {JdW}}\right| \geq \varepsilon }\right\} \]\n\nthen\n\n\[ ... | Yes |
Lemma 73.6.5 Let \( X\left( s\right) - {X}_{k}^{l}\left( s\right) \equiv {\Delta }_{k}\left( s\right) \) . Then the following limit occurs.\n\n\[ \mathop{\lim }\limits_{{k \rightarrow \infty }}P\left( \left\lbrack {\mathop{\sup }\limits_{{t \in \left\lbrack {0, T}\right\rbrack }}\left| {{\int }_{0}^{t}{\left( Z \circ {... | Proof: This follows from Lemma 73.6.3. The last conclusion follows from the usual use of the Borel Cantelli lemma. There exists a further subsequence, still denoted with subscript \( k \) such that\n\n\[ P\left( \left\lbrack {\mathop{\sup }\limits_{{t \in \left\lbrack {0, T}\right\rbrack }}\left| {{\int }_{0}^{t}{\left... | Yes |
Theorem 73.7.2 In Situation 73.2.1, for \( \omega \) off a set of measure zero, for every \( t \notin {N}_{\omega }, \)\n\n\[ \langle {BX}\left( t\right), X\left( t\right) \rangle = \left\langle {B{X}_{0},{X}_{0}}\right\rangle + {\int }_{0}^{t}\left( {2\langle Y\left( s\right), X\left( s\right) \rangle +\langle {BZ}, Z... | Proof: Let \( t \in {N}_{\omega }^{C} \smallsetminus D \) . For \( t > 0 \), let \( t\left( k\right) \) denote the largest point of \( {\mathcal{P}}_{k} \) which is less than \( t \) . Suppose \( t\left( m\right) < t\left( k\right) \) . Hence \( m \leq k \) . Then\n\n\[ {BX}\left( {t\left( m\right) }\right) = B{X}_{0} ... | Yes |
Lemma 74.0.1 Let \( {f}_{n} \rightarrow f \) in \( {L}^{p}\left( {\left\lbrack {0, T}\right\rbrack \times \Omega, E}\right) \). Then there exists a subsequence \( {n}_{k} \) and a set of measure zero \( N \) such that if \( \omega \notin N \), then\n\n\[ \n{f}_{{n}_{k}}\left( {\cdot ,\omega }\right) \rightarrow f\left(... | Proof: We have\n\n\[ \nP\left( \left\lbrack {{\begin{Vmatrix}{f}_{n} - f\end{Vmatrix}}_{{L}^{p}\left( {\left\lbrack {0, T}\right\rbrack, E}\right) } > \lambda }\right\rbrack \right) \leq \frac{1}{\lambda }{\int }_{\Omega }{\begin{Vmatrix}{f}_{n} - f\end{Vmatrix}}_{{L}^{p}\left( {\left\lbrack {0, T}\right\rbrack, E}\rig... | Yes |
Lemma 74.0.2 Let \( \Phi : \left\lbrack {0, T}\right\rbrack \times \Omega \rightarrow V \), be \( \mathcal{B}\left( \left\lbrack {0, T}\right\rbrack \right) \times \mathcal{F} \) measurable and suppose\n\n\[ \Phi \in K \equiv {L}^{p}\left( {\left\lbrack {0, T}\right\rbrack \times \Omega ;E}\right), p \geq 1 \]\n\nThen ... | Also, each \( \Phi \left( {t}_{j}^{k}\right) ,\Phi \left( {t}_{j - 1}^{k}\right) \) is in \( {L}^{p}\left( {\Omega ;E}\right) \) . One can also assume that \( \Phi \left( 0\right) = 0 \) . The mesh points \( {\left\{ {t}_{j}^{k}\right\} }_{j = 0}^{{m}_{k}} \) can be chosen to miss a given set of measure zero. In additi... | Yes |
Lemma 74.2.1 In the above situation, there exists a set of measure zero \( N \subseteq \Omega \) and a dense subset of \( \left\lbrack {0, T}\right\rbrack, D \) such that for \( \omega \notin N,{BX}\left( {t,\omega }\right) = B\left( {X\left( {t,\omega }\right) }\right) \) for all \( t \in D \) . This set \( D \) is th... | \[ \langle {BX}\left( t\right) \left( \omega \right), X\left( {t,\omega }\right) \rangle = \mathop{\sum }\limits_{i}{\left\langle B\left( X\left( t\right) \right) ,{e}_{i}\right\rangle }^{2}\text{ a.e. }\omega . \] \( D \) has empty intersection with \( \widehat{N} \) . There is also a set of Lebesgue measure zero \( {... | Yes |
Theorem 74.2.2 Let \( {\left\{ {t}_{j}^{n}\right\} }_{j = 0}^{{m}_{n}} \) be the above sequence of partitions of the sort in Lemma 74.0.2 such that if\n\n\[ \n{X}_{n}^{l}\left( t\right) \equiv \mathop{\sum }\limits_{{j = 0}}^{{{m}_{n} - 1}}X\left( {t}_{j}^{n}\right) {\mathcal{X}}_{\left\lbrack {t}_{j}^{n},{t}_{j + 1}^{... | Proof: This follows from Lemma 66.0.20. This can be seen because, thanks to the fact that \( B{X}_{k}^{l{\sigma }_{q}^{k}} \) is bounded, the function \( B{X}_{k}^{l} \) is in the set \( \mathcal{G} \) described there. This is a place where we use that \( d\left\lbrack M\right\rbrack = {kdt} \) . | Yes |
Lemma 74.3.1 In Situation 74.1.1 the following formula holds for a.e. \( \omega \) for \( 0 < \) \( s < t \) . In the following, \( \langle \cdot , \cdot \rangle \) denotes the duality pairing between \( V,{V}^{\prime } \) .\n\n\[ \langle {BX}\left( t\right), X\left( t\right) \rangle = \langle {BX}\left( s\right), X\le... | Proof: From the formula which is assumed to hold,\n\n\[ {BX}\left( t\right) = B{X}_{0} + {\int }_{0}^{t}Y\left( u\right) {du} + {BM}\left( t\right) \]\n\n\[ {BX}\left( s\right) = B{X}_{0} + {\int }_{0}^{s}Y\left( u\right) {du} + {BM}\left( s\right) \]\n\nThen\n\n\[ {BM}\left( t\right) - {BM}\left( s\right) + {\int }_{s... | Yes |
In the Situation 74.1.1, the following holds for all \( t \notin \widehat{N} \): \n\n\[ \nE\left( {\langle {BX}\left( t\right), X\left( t\right) \rangle }\right) \n\] \n\n\[ \n< C\left( {\parallel Y{\parallel }_{{K}^{\prime }},\parallel X{\parallel }_{K}, E\left( {\left\lbrack M\right\rbrack \left( T\right) }\right) ,{... | Proof: Consider the formula in Lemma 74.3.1. \n\n\[ \n\langle {BX}\left( t\right), X\left( t\right) \rangle = \langle {BX}\left( s\right), X\left( s\right) \rangle \n\] \n\n\[ \n+ 2{\int }_{s}^{t}\langle Y\left( u\right), X\left( t\right) \rangle {du} + \langle B\left( {M\left( t\right) - M\left( s\right) }\right), M\l... | Yes |
Theorem 74.6.2 In Situation 74.1.1, for \( \omega \) off a set of measure zero, it follows that for every \( t \in {N}_{\omega }^{C} \) ,\n\n\[ \langle {BX}\left( t\right), X\left( t\right) \rangle = \left\langle {B{X}_{0},{X}_{0}}\right\rangle + {\int }_{0}^{t}2\langle Y\left( s\right), X\left( s\right) \rangle {ds} \... | Proof: Let \( t \in {N}_{\omega }^{C} \smallsetminus D \) . For \( t > 0 \), let \( t\left( k\right) \) denote the largest point of \( {\mathcal{P}}_{k} \) which is less than \( t \) . Suppose \( t\left( m\right) < t\left( k\right) \) . Hence \( m \leq k \) . Then\n\n\[ {BX}\left( {t\left( m\right) }\right) = B{X}_{0} ... | Yes |
Theorem 74.6.3 In Situation 74.1.1 in which \( W = H = {H}^{\prime } \) and \( B = I \), it follows that off a set of measure zero, for every \( t \in \left\lbrack {0, T}\right\rbrack \), there is a set of measure zero \( N \) such that for \( \omega \notin N \), there is a continuous function \( \langle X, X\rangle \)... | \[ \langle X, X\rangle \left( t\right) = {\left| {X}_{0}\right| }_{H}^{2} + {\int }_{0}^{t}2\langle Y\left( s\right), X\left( s\right) \rangle {ds} + \left\lbrack M\right\rbrack \left( t\right) + 2{\int }_{0}^{t}\left( {X,{dM}}\right) \] | Yes |
Theorem 75.1.3 Let \( V \) be a Banach space and let \( A : V \rightarrow {V}^{\prime } \) be monotone and hemicontinuous. Then \( A \) is pseudomonotone. | Proof: Let \( A \) be monotone and Hemicontinuous. First here is a claim.\n\nClaim: If 75.1.1 and 75.1.2 hold, then \( \mathop{\lim }\limits_{{n \rightarrow \infty }}\left\langle {A{u}_{n},{u}_{n} - u}\right\rangle = 0 \) .\n\nProof of the claim: Since \( A \) is monotone,\n\n\[ \left\langle {A{u}_{n} - {Au},{u}_{n} - ... | Yes |
Example 75.1.4 Let \( H \) be any Hilbert space and let \( A : H \rightarrow {H}^{\prime } \) be given by\n\n\[ \langle {Ax}, y\rangle \equiv {\left( -x, y\right) }_{H}. \]\n\nThen \( A \) fails to be pseudomonotone. | Proof: Let \( {\left\{ {x}_{n}\right\} }_{n = 1}^{\infty } \) be an orthonormal set of vectors in \( H \) . Then Parsevall’s inequality implies\n\n\[ \parallel x{\parallel }^{2} \geq \mathop{\sum }\limits_{{n = 1}}^{\infty }{\left| \left( {x}_{n}, x\right) \right| }^{2} \]\n\nand so for any \( x \in H,\mathop{\lim }\li... | Yes |
Proposition 75.1.5 Suppose \( A : V \rightarrow {V}^{\prime } \) is pseudomonotone and bounded where \( V \) is separable. Then it must be demicontinuous. This means that if \( {u}_{n} \rightarrow u \), then \( A{u}_{n} \rightharpoonup {Au} \) . | Proof: Since \( {u}_{n} \rightarrow u \) is strong convergence and since \( A{u}_{n} \) is bounded, it follows\n\n\[ \lim \mathop{\sup }\limits_{{n \rightarrow \infty }}\left\langle {A{u}_{n},{u}_{n} - u}\right\rangle = \mathop{\lim }\limits_{{n \rightarrow \infty }}\left\langle {A{u}_{n},{u}_{n} - u}\right\rangle = 0.... | Yes |
Proposition 75.1.7 If \( A \) is pseudomonotone, then \( A \) is type \( M \) . | Proof: Suppose \( A \) is pseudomonotone and \( {u}_{n} \rightharpoonup u \) and \( A{u}_{n} \rightharpoonup \xi \), and\n\n\[ \lim \mathop{\sup }\limits_{{n \rightarrow \infty }}\left\langle {A{u}_{n},{u}_{n}}\right\rangle \leq \langle \xi, u\rangle \]\n\nThen\n\n\[ \lim \mathop{\sup }\limits_{{n \rightarrow \infty }}... | Yes |
Proposition 75.1.8 Suppose \( A : V \rightarrow {V}^{\prime } \) is type \( M \) and suppose \( L : V \rightarrow {V}^{\prime } \) is monotone, bounded and linear. Then \( L + A \) is type \( M \) . Let \( V \) be separable or reflexive so that the weak convergences in the following argument are valid. | Proof: Suppose \( {u}_{n} \rightharpoonup u \) and \( A{u}_{n} + L{u}_{n} \rightharpoonup \xi \) and also that\n\n\[ \lim \mathop{\sup }\limits_{{n \rightarrow \infty }}\left\langle {A{u}_{n} + L{u}_{n},{u}_{n}}\right\rangle \leq \langle \xi, u\rangle \]\n\nDoes it follow that \( \xi = {Au} + {Lu} \) ? Suppose not. The... | Yes |
Corollary 75.1.9 Suppose \( A : V \rightarrow {V}^{\prime } \) is type \( M \) and suppose \( L : V \rightarrow {V}^{\prime } \) is monotone, bounded and linear. Then for \( {u}_{0} \in V \) define \( M\left( u\right) \equiv L\left( {u - {u}_{0}}\right) \) . Then \( M + A \) is type \( M \) . Let \( V \) be separable o... | Proof: Suppose \( {u}_{n} \rightharpoonup u \) and \( A{u}_{n} + M{u}_{n} \rightharpoonup \xi \) and also that\n\n\[ \lim \mathop{\sup }\limits_{{n \rightarrow \infty }}\left\langle {A{u}_{n} + M{u}_{n},{u}_{n}}\right\rangle \leq \langle \xi, u\rangle \]\n\nDoes it follow that \( \xi = {Au} + {Mu} \) ? Suppose not. By ... | Yes |
Lemma 75.1.10 (Browder) Let \( K \) be a convex closed and bounded set in \( {\mathbb{R}}^{n} \) and let \( A : K \rightarrow {\mathbb{R}}^{n} \) be continuous and \( \mathbf{f} \in {\mathbb{R}}^{n} \) . Then there exists \( \mathbf{x} \in K \) such that for all \( \mathbf{y} \in K \) , \[ \left( {\mathbf{f} - A\mathbf... | Proof: Let \( {P}_{K} \) denote the projection onto \( K \) . Thus \( {P}_{K} \) is Lipschitz continuous. \[ \mathbf{x} \rightarrow {P}_{K}\left( {\mathbf{f} - A\mathbf{x} + \mathbf{x}}\right) \] is a continuous map from \( K \) to \( K \) . By the Brouwer fixed point theorem, it has a fixed point \( \mathbf{x} \in K \... | Yes |
Proposition 75.1.11 Let \( A : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) be continuous and coercive,\n\n\[ \mathop{\lim }\limits_{{\left| \mathbf{x}\right| \rightarrow \infty }}\frac{\left( A\left( \mathbf{x} + {\mathbf{x}}_{0}\right) ,\mathbf{x}\right) }{\left| \mathbf{x}\right| } = \infty \]\n\nfor some \( {\m... | Proof: Define the closed convex sets \( {B}_{n} \equiv \overline{B\left( {{\mathbf{x}}_{0}, n}\right) } \) . By Browder’s lemma, there exists \( {\mathbf{x}}_{n} \) such that\n\n\[ \left( {\mathbf{f} - A{\mathbf{x}}_{n},\mathbf{y} - {\mathbf{x}}_{n}}\right) \leq 0 \]\n\nfor all \( \mathbf{y} \in {B}_{n} \) . Then takin... | Yes |
Lemma 75.1.12 Let \( A : V \rightarrow {V}^{\prime } \) be type \( M \) and bounded and suppose \( V \) is reflexive or \( V \) is separable. Then \( A \) is demicontinuous. | Proof: Suppose \( {u}_{n} \rightarrow u \) and \( A{u}_{n} \) fails to converge weakly to \( {Au} \) . Then there is a further subsequence, still denoted as \( {u}_{n} \) such that \( A{u}_{n} \rightharpoonup \zeta \neq {Au} \) . Then thanks to the strong convergence, you have\n\n\[ \lim \mathop{\sup }\limits_{{n \righ... | No |
Lemma 75.2.2 Let \( F \) be a duality map for \( p = 2 \) where \( X,{X}^{\prime } \) are reflexive and have strictly convex norms. (If \( X \) is reflexive, there is always an equivalent strictly convex norm [8].) Then \( F \) is demicontinuous. | Proof: Say \( {x}_{n} \rightarrow x \) . Then does it follow that \( F{x}_{n} \rightharpoonup {Fx} \) ? Suppose not. Then there is a subsequence, still denoted as \( {x}_{n} \) such that \( {x}_{n} \rightarrow x \) but \( F{x}_{n} \rightharpoonup y \neq {Fx} \) where here \( \rightharpoonup \) denotes weak convergence.... | Yes |
Lemma 75.2.4 Let \( p > 2 \) . Then for \( a, b \) real numbers, \( \left( {{\left| a\right| }^{p - 2}a - {\left| b\right| }^{p - 2}b}\right) \left( {a - b}\right) \geq \) \( C{\left| a - b\right| }^{p} \) for some constant \( C \) independent of \( a, b \) . | Proof: There is nothing to show if \( a = b \) . Without loss of generality, assume \( a > b \) . Also assume \( p \geq 2 \) . There is nothing to show if \( p = 2 \) . I want to show that there exists a constant \( C \) such that for \( a > b \) ,\n\n\[ \frac{{\left| a\right| }^{p - 2}a - {\left| b\right| }^{p - 2}b}{... | Yes |
Theorem 75.2.5 Let \( X \) be a reflexive Banach space and \( X,{X}^{\prime } \) have strictly convex norms as discussed above. Let \( F \) be the duality map with \( p = 2 \) . Then \( F \) is strictly monotone. This means\n\n\[ \langle {Fu} - {Fv}, u - v\rangle \geq 0 \]\n\nand it equals 0 if and only if \( u - v \) ... | Proof: First why is it monotone? By definition of \( F,\langle F\left( u\right), u\rangle = \parallel u{\parallel }^{2} \) and \( \parallel F\left( u\right) \parallel = \parallel u\parallel \) . Then\n\n\[ \left| {\langle {Fu}, v\rangle }\right| = \left| \left\langle {{Fu},\frac{v}{\parallel v\parallel }}\right\rangle ... | Yes |
Theorem 76.2.3 Off a set of measure zero, for every \( t \in \left\lbrack {0, T}\right\rbrack \) , \n\n\[ \n\langle {BX}, X\rangle \left( t\right) = \left\langle {B{X}_{0},{X}_{0}}\right\rangle + {\int }_{0}^{t}\left( {2\langle Y\left( s\right), X\left( s\right) \rangle +\langle {BZ}, Z{\rangle }_{{\mathcal{L}}_{2}}}\r... | Also \n\n\[ \nE\left( {\langle {BX}, X\rangle \left( t\right) }\right) = \n\] \n\n\[ \nE\left( \left\langle {B{X}_{0},{X}_{0}}\right\rangle \right) + E\left( {{\int }_{0}^{t}\left( {2\langle Y\left( s\right), X\left( s\right) \rangle +\langle {BZ}, Z{\rangle }_{{\mathcal{L}}_{2}}}\right) {ds}}\right) \n\] \n\n\( \left(... | No |
Theorem 76.2.8 Suppose that \( f \) and \( {Df} \) are both in \( {L}^{1}\left( {a, b,{V}^{\prime }}\right) \) . Then \( f \) is equal to a continuous function a.e., still denoted by \( f \) and | \[ f\left( x\right) = f\left( a\right) + {\int }_{a}^{x}{Df}\left( t\right) {dt}. \] | Yes |
Lemma 76.3.3 Suppose it is true that whenever \( u, v \in {\mathcal{V}}_{\omega } \) and\n\n\[ \n{Bu}\left( t\right) - {Bv}\left( t\right) + {\int }_{0}^{t}A\left( u\right) - A\left( v\right) = 0 \n\]\n\n\( \left( {76.3.16}\right) \)\n\nit follows that \( u = v \) . Then if\n\n\[ \n{\left( Bu\right) }^{\prime } + \bar{... | Proof: If \( {\left( Bu\right) }^{\prime } + \bar{A}\left( \omega \right) u = f \) and \( {\left( Bv\right) }^{\prime } + \bar{A}\left( \omega \right) v = f \), then\n\n\[ \n{Bu}\left( t\right) - {Bv}\left( t\right) + {\int }_{0}^{t}A\left( {u + q}\right) - A\left( {v + q}\right) {ds} = 0 \n\]\n\nHence\n\n\[ \nB\left( ... | Yes |
Lemma 76.3.5 Let \( q \in \mathcal{V} \) and let the conditions 76.3.14 - 76.3.17 be valid. Let \( f \in {\mathcal{V}}^{\prime } \) be given. Then for each \( \omega \) off a set of measure zero, there exists \( u\left( {\cdot ,\omega }\right) \in {\mathcal{V}}_{\omega } \) such that \( {\left( Bu\right) }^{\prime }\le... | Proof: Consider the equation\n\n\[ \n{L}_{h}{Bu} + \bar{A}u = \frac{1}{h}\left( {I - {\tau }_{h}}\right) \left( {Bu}\right) + \bar{A}u = f\text{ in }{\mathcal{V}}^{\prime } \n\]\n\n(76.3.18)\n\nBy Proposition 76.2.4 and Theorem 76.2.6, there exists a solution to the above equation if the left side is coercive. However,... | Yes |
Proposition 76.3.6 Let \( q \in \mathcal{V} \) such that \( t \rightarrow q\left( {t,\omega }\right) \) is continuous and \( q\left( {0,\omega }\right) = 0 \) , and let the conditions 76.3.14 - 76.3.17 be valid. Also let \( {u}_{0} \in {L}^{2}\left( {\Omega, V}\right) \) such that \( {u}_{0} \) is \( {\mathcal{F}}_{0} ... | Proof: Recall\n\n\[ \n\bar{A}\left( \omega \right) \left( {t, u}\right) \equiv A\left( {t, u + q\left( {t,\omega }\right) ,\omega }\right) \]\n\nwhere \( q \) was in \( \mathcal{V} \) . Therefore, replace this definition of \( \bar{A} \) with\n\n\[ \n\bar{A}\left( \omega \right) \left( {t, u}\right) \equiv A\left( {t, ... | Yes |
Corollary 76.3.7 Suppose the situation of the above proposition but that all that is known is that \( {\lambda B} + A \) is monotone and hemicontinuous on \( {\mathcal{V}}_{\omega } \) and \( \mathcal{V} \) for all \( \lambda \) sufficiently large. Then defining\n\n\[ \n{\left\langle {A}_{\lambda }\left( t, w,\omega \r... | Proof: That \( {\lambda B} + {A}_{\lambda } \) is monotone and hemicontinuous follows from the definition. Also, from the above estimates,\n\n\[ \n\lambda \left\langle {{Bu}, u}\right\rangle + {\left\langle {A}_{\lambda }\left( t, u,\omega \right), u\right\rangle }_{V} \geq {e}^{-{2\lambda t}}\left( {\lambda \left\lang... | Yes |
Lemma 76.4.1 Suppose \( {u}_{n} \rightarrow w \) weakly in \( {\mathcal{V}}_{\omega } \) and that for a.e.t, \( {u}_{n}\left( t\right) \rightarrow u\left( t\right) \) in U. Then \( w\left( t\right) = u\left( t\right) \) a.e. | Proof: You know that \( {\begin{Vmatrix}{u}_{n}\end{Vmatrix}}_{{L}^{p}\left( {\left\lbrack {0, T}\right\rbrack, V}\right) } \) is bounded. Now consider \( \phi \in {U}^{\prime } \) and \( \psi \in C\left( \left\lbrack {0, T}\right\rbrack \right) \) . Then the weak convergence implies\n\n\[ \mathop{\lim }\limits_{{n \ri... | Yes |
Lemma 76.4.2 There is a subsequence, still indexed by \( n \) and a set of measure zero \( N \), containing all the preceding sets of measure zero such that for \( \omega \notin N \), \[ \mathop{\sup }\limits_{{s \in \left\lbrack {0, T}\right\rbrack }}\left\langle {B{u}_{n},{u}_{n}}\right\rangle \left( s\right) + {\int... | From the theory of the stochastic integral, there is a further subsequence of the above such that \[ {\int }_{0}^{t}{\Phi }_{n}{dW} \rightarrow {\int }_{0}^{t}{\Phi dW}\text{ strongly in }C\left( {\left\lbrack {0, T}\right\rbrack, W}\right) \] for all \( \omega \) off a set of measure zero. Enlarge the exceptional set ... | Yes |
Lemma 76.4.3 The above sequence does not depend on \( \omega \notin N \) . In fact, it is not necessary to take a further subsequence. | Proof: In fact, it is not necessary to take a subsequence to get the convergences 76.4.38 - 76.4.40. This is because of the pointwise convergence of 76.4.30 and Lemma 76.4.1. If the original sequence did not converge, then there would be two subsequences converging weakly to two different functions in \( {\mathcal{V}}_... | Yes |
Lemma 76.4.4 Enlarging the exceptional set, one can assume that \( \xi \) is also progressively measurable. In fact, if\n\n\[ \n{\xi }_{n} \equiv {\int }_{t - \left( {1/n}\right) }^{t}{\xi ds} \n\]\n\nis known to be progressively measurable, \( \xi \left( {t,\omega }\right) \equiv 0 \) for \( t < 0 \), then there exist... | Proof: Define\n\n\[ \n{\xi }_{n} \equiv n{\int }_{t - \left( {1/n}\right) }^{t}{\xi ds} \n\]\n\nwhere \( \xi \) is defined to be zero for \( t \leq 0 \) . Then by what was just shown, this is progressively measurable. Also, standard approximate identity arguments verify that for each \( \omega ,{\xi }_{n} \rightarrow \... | Yes |
Lemma 76.4.5 It is true that\n\n\\[ \n\\mathop{\\lim }\\limits_{{n \\rightarrow \\infty }}{\\int }_{0}^{T}\\left\\langle {B{u}_{n},{u}_{n}}\\right\\rangle {dt} = {\\int }_{0}^{T}\\langle {Bu}, u\\rangle {dt} \n\\] | Proof: From 76.4.28 \\( B{u}_{n} \\rightarrow z \\) strongly in \\( C\\left( {{N}_{\\omega }^{C},{W}^{\\prime }}\\right) \\) . But also, for each \\( t, B{u}_{n}\\left( t\\right) \\rightarrow {Bu}\\left( t\\right) \\) weakly in \\( {V}^{\\prime } \\) and so \\( z\\left( t\\right) = {Bu}\\left( t\\right) \\) . This stro... | Yes |
Theorem 76.4.7 Suppose \( \mathcal{V} \equiv {L}^{p}\left( {\left\lbrack {0, T}\right\rbrack \times \Omega, V}\right) \) where \( p \geq 2 \), with the \( \sigma \) algebra of progressively measurable sets and \( {\mathcal{V}}_{\omega } = {L}^{p}\left( {\left\lbrack {0, T}\right\rbrack, V}\right) \) . \n\n\[ \Phi \in {... | Proof: The uniqueness assertion follows easily from the monotonicity condition. \( ▱ \) | No |
Theorem 76.4.9 Suppose \( \mathcal{V} \equiv {L}^{p}\left( {\left\lbrack {0, T}\right\rbrack \times \Omega, V}\right) \) where \( p \geq 2 \), with the \( \sigma \) algebra of progressively measurable sets and \( {\mathcal{V}}_{\omega } = {L}^{p}\left( {\left\lbrack {0, T}\right\rbrack, V}\right) \). | \[ \Phi \in {L}^{2}\left( {\left\lbrack {0, T}\right\rbrack \times \Omega ,{\mathcal{L}}_{2}\left( {{Q}^{1/2}U, W}\right) }\right) \] \[ f \in {\mathcal{V}}^{\prime } \equiv {L}^{{p}^{\prime }}\left( {\left\lbrack {0, T}\right\rbrack \times \Omega ,{V}^{\prime }}\right) \] and both are progressively measurable. Suppose... | Yes |
Example 76.6.1 The stochastic porous media equation is\n\n\[ \n{u}_{t} - \Delta \left( {u{\left| u\right| }^{p - 2}}\right) = f, u\left( 0\right) = {u}_{0}, u = 0\text{ on }\partial U \n\] \n\nwhere here \( U \) is a bounded open set in \( {\mathbb{R}}^{n}, n \leq 3 \) having Lipschitz boundary. One can consider a stoc... | One can consider this as an implicit integral equation of the form\n\n\[ \n{\left( -\Delta \right) }^{-1}u\left( t\right) - {\left( -\Delta \right) }^{-1}{u}_{0} + {\int }_{0}^{t}u{\left| u\right| }^{p - 2}{ds} = {\left( -\Delta \right) }^{-1}{\int }_{0}^{t}{\Phi dW} + {\left( -\Delta \right) }^{-1}{\int }_{0}^{t}{fds}... | Yes |
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