## The Annals of Probability

### 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.

#### Article information

Source
Ann. Probab., Volume 42, Number 5 (2014), 2113-2160.

Dates
First available in Project Euclid: 29 August 2014

https://projecteuclid.org/euclid.aop/1409319474

Digital Object Identifier
doi:10.1214/14-AOP936

Mathematical Reviews number (MathSciNet)
MR3262499

Zentralblatt MATH identifier
1328.35291

#### Citation

Da Prato, Giuseppe; Lunardi, Alessandra. Sobolev regularity for a class of second order elliptic PDE’s in infinite dimension. Ann. Probab. 42 (2014), no. 5, 2113--2160. doi:10.1214/14-AOP936. https://projecteuclid.org/euclid.aop/1409319474

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