Differentiability of Markov semigroups for stochastic reaction-diffusion equations and applications to control
type: Item
instance of: scholarly article
title: Differentiability of Markov semigroups for stochastic reaction-diffusion equations and applications to control
zbMATH Open document ID: 0996.60072
DOI: 10.1016/S0304-4149(99)00014-9
author: Sandra Cerrai
published in: Stochastic Processes and their Applications
publication date: 2002-08-29T00:00:00Z
review text: The author studies smoothing properties of a transition semigroup corresponding to a semilinear stochastic reaction-diffusion equation with an additive noise, strengthening known results by allowing a polynomial growth of the nonlinear term and considering diffusion operators which are not boundedly invertible. The regularity results are applied to obtain solutions to certain infinite-dimensional Hamilton-Jacobi equations. Let us state the main result of the paper (Theorem 5.1) more precisely: Let \(U\subseteq \mathbb R^{d}\) be a bounded domain with a regular boundary. Set \(H = L^2(U)\), \(E= C(\overline U)\), an equation \[ du(t) = \bigl \{ Au(t) + F(u(t))\bigr \} dt + B dw \tag{1} \] in \(H\) is considered, \(w\) being a standard cylindrical Wiener process on \(H\). Suppose that \(A\) is a negative operator generating an analytic self-adjoint \(C_0\)-semigroup \((e^{At})\) on \(H\). Let \((e^{At})\) have an extension on \(L^{p}(U)\) such that \(e^{At}\) maps \(L^{p}(U)\) into the Sobolev-Slobodetskij space \(W^{s,p}(U)\) for all \(p\geq 1\), \(s\in ]0,1[\) and \(t>0\), and \(|e^{At}x|_{W^{s,p}}\leq ct^{-s/r}|x|_{L^{p}}\) for some \(r>d/2\). Let there exist an orthonormal basis \(\{e_{k}\}\) of \(H\) consisting of eigenvectors of the operator \(A\), \(Ae_{k} = -\alpha _{k} e_{k}\), and \(\sup _{k} |e_{k}|_{E} <\infty \), \(\sup _{\xi \in \overline U}|\nabla e_{k}(\xi)|\leq c\alpha ^{1/2}_{k}\). Further, let the part assumed to be bounded, nonnegative, self-adjoint and diagonal with respect to the basis \(\{e_{k}\}\), \(Be_{k} = \lambda _{k}e_{k}\). Let \(\sum ^{\infty }_{k=1} \alpha ^{\gamma -1}_{k}\lambda ^2_{k} <\infty \) for a \(\gamma \in ]0,1[\). Moreover, suppose that \(\text{Dom} ((-A)^{\varepsilon /2})\subseteq \text{Rng } B\) for an \(\varepsilon <1\). The nonlinear term \(F\) is defined as a Nemytskij operator given by a function \(f\in C^2(\mathbb R)\) of a polynomial growth, \(\sup _{t\in \mathbb R} (1+|t|^{2m+1-j})^{-1} |f^{(j)}(t)|<\infty \) for an …
reviewed by: Jan Seidler
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60J35
zbMATH DE Number: 1792528
zbMATH Keywords: stochastic reaction-diffusion equations
zbMATH Keywords: Bismut-Elworthy formula
zbMATH Keywords: Markov semigroups
zbMATH Keywords: Hamilton-Jacobi equations
MaRDI profile type: Publication
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full work available at URL: s0304-4149(99)00014-9
OpenAlex ID: W1994682102
description: scientific article; zbMATH DE number 1792528