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September 2014 Sobolev regularity for a class of second order elliptic PDE’s in infinite dimension
Giuseppe Da Prato, Alessandra Lunardi
Ann. Probab. 42(5): 2113-2160 (September 2014). DOI: 10.1214/14-AOP936


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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Giuseppe Da Prato. Alessandra Lunardi. "Sobolev regularity for a class of second order elliptic PDE’s in infinite dimension." Ann. Probab. 42 (5) 2113 - 2160, September 2014.


Published: September 2014
First available in Project Euclid: 29 August 2014

zbMATH: 1328.35291
MathSciNet: MR3262499
Digital Object Identifier: 10.1214/14-AOP936

Primary: 35B65 , 35R15 , 37L40

Keywords: Invariant measures , Kolmogorov operators in infinite dimensions , Maximal Sobolev regularity

Rights: Copyright © 2014 Institute of Mathematical Statistics


Vol.42 • No. 5 • September 2014
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