Differentiability of Markov semigroups for stochastic reaction-diffusion equations and applications to control
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 \(m\geq 1\) and \(j=0,1,2\). Let \(\sup f^\prime <\infty \). Finally, a dissipativity assumption \(f(\sigma +\rho)-f(\rho)\leq -a\sigma ^{2m+1} + b|\rho |^{2m+1} + c\) for a positive constant \(a\), some real constants \(b,c\) and every \(\sigma >0\), \(\rho \in \mathbb R\), is adopted. Denote by \((P_{t})\) the transition semigroup of the Markov process solving (1). Under the above hypotheses, \(P_{t}\psi \) is a bounded uniformly continuous function on \(H\), having a bounded uniformly continuous Fréchet derivative, for all bounded Borel functions \(\psi \) on \(H\) and each \(t>0\). Furthermore, the Bismut-Elworthy formula holds for the directional derivatives of \(P_{t}\psi \).
scientific article; zbMATH DE number 1792528