A relative compactness criterion in Wiener-Sobolev spaces and application to semi-linear stochastic PDEs
Let \(O\subset \mathbb R^{d}\) be a bounded domain. The authors consider a sequence of stochastic processes \(u_{n} :[0,T]\times O\times\Omega\to \mathbb R\), \(n\geq1\), bounded in \(L^{2}([0,T]\times\Omega,H^{1}(O))\), and such that, for every \(\varphi\in C_{c}^{\infty}(\mathbb R^{d})\), the functions \(u_{n}^{\varphi} (t):=\int\varphi(x)u_{n}(t,x)\,dx\), \(n\geq1\), are Malliavin differentiable (with Malliavin derivative \(D_{\theta}u_{n}^{\varphi}(t)\)), bounded in \(L^{2}([0,T];D^{1,2})\) and have the property that \(t\to E[u_{n} ^{\varphi}(t)]\) and \(( t,\theta) \rightarrow E[D_{\theta} u_{n}^{\varphi}(t)]\) are continuous in \(L^{2}([0,T])\) and \(L^{2}([0,T]^{2})\), respectively. They prove that under these hypotheses the sequence \((u_{n})_{n\geq1}\) is relatively compact in \(L^{2}([0,T]\times O \times\Omega)\). This relative compactness criterion represents a natural extension of the compact Sobolov embedding of the space of deterministic functions \(H^{1}(O)\) into \(L^{2}(O)\). After the authors apply this criterion to semi-linear stochastic PDEs in order to describe classes of coefficients for which the set of associated solutions \(u\) form a relatively compact subset of \(L^{2}([0,\tau]\times O\times\Omega)\), for all \(\tau<T\). For the proof the authors use the stochastic interpretation of the solutions \(u\) by backward doubly SDEs. Finally the authors apply their result on the relative compactness of classes of solutions of stochastic PDEs to study the existence and the uniqueness of semi-linear stochastic PDEs with a distribution as final condition. This distribution is supposed to belong to the dual of a weighted Sobolev space defined as the completion of \(C_{c} ^{k}(R^{d})\), where \(k\geq1\). The authors also give a stochastic interpretation of the solution of such equations in terms of backward doubly stochastic differential equations formulated in a weak sense.
stochastic partial differential equations
backward doubly stochastic differental equations
scientific article; zbMATH DE number 2091727