Maximal $L^2$ regularity for Ornstein-Uhlenbeck equation in convex sets of Banach spaces

TL;DR

This paper establishes maximal $L^2$ regularity for solutions to the Ornstein-Uhlenbeck equation in convex subsets of infinite-dimensional Banach spaces with Gaussian measures, including boundary trace properties.

Contribution

It proves that solutions in this setting belong to the Sobolev space $W^{2,2}$ and satisfy Neumann boundary conditions via finite dimensional approximation.

Findings

Solutions are in $W^{2,2}( ext{Omega}, ext{gamma})$ for $ ext{lambda}>0$ and $f ext{ in }L^2( ext{Omega}, ext{gamma})$.

Solutions satisfy Neumann boundary conditions in the trace sense.

The results are obtained through finite dimensional approximation techniques.

Abstract

We study the elliptic equation $\lambda u-L^{\Omega}u=f$ in an open convex subset $\Omega$ of an infinite dimensional separable Banach space $X$ endowed with a centered non-degenerate Gaussian measure $\gamma$, where $L^\Omega$ is the Ornstein-Uhlenbeck operator. We prove that for $\lambda>0$ and $f\in L^2(\Omega,\gamma)$ the weak solution $u$ belongs to the Sobolev space $W^{2,2}(\Omega,\gamma)$. Moreover we prove that $u$ satisfies the Neumann boundary condition in the sense of traces at the boundary of $\Omega$. This is done by finite dimensional approximation.