Sobolev regularity for a class of second order elliptic PDE’s in infinite dimension

Abstract

We consider an elliptic Kolmogorov equation $\lambda u-Ku=f$ in a separable Hilbert space $H$. The Kolmogorov operator $K$ is associated to an infinite dimensional convex gradient system: $dX=(AX-DU(X))\,dt+dW(t)$, where $A$ is a self-adjoint operator in $H$, and $U$ is a convex lower semicontinuous function. Under mild assumptions we prove that for $\lambda>0$ and $f\in L^{2}(H,\nu)$ the weak solution $u$ belongs to the Sobolev space $W^{2,2}(H,\nu)$, where $\nu$ is the log-concave probability measure of the system. Moreover maximal estimates on the gradient of $u$ are proved. The maximal regularity results are used in the study of perturbed nongradient systems, for which we prove that there exists an invariant measure. The general results are applied to Kolmogorov equations associated to reaction–diffusion and Cahn–Hilliard stochastic PDEs.