The Annals of Probability

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

Giuseppe Da Prato and Alessandra Lunardi

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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.

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

First available in Project Euclid: 29 August 2014

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Primary: 35R15: Partial differential equations on infinite-dimensional (e.g. function) spaces (= PDE in infinitely many variables) [See also 46Gxx, 58D25] 37L40: Invariant measures 35B65: Smoothness and regularity of solutions

Kolmogorov operators in infinite dimensions maximal Sobolev regularity invariant measures


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.

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