Infinite dimensional reflecting Ornstein-Uhlenbeck stochastic process

TL;DR

This paper develops the Gaussian Sobolev space framework in infinite dimensions and studies the semimartingale properties of the reflecting Ornstein-Uhlenbeck process in such settings.

Contribution

It introduces the Gaussian Sobolev space on open sets in infinite-dimensional Banach spaces and analyzes the semimartingale structure of the reflecting Ornstein-Uhlenbeck process.

Findings

Defined Gaussian Sobolev space $W^{1,2}( abla)$ in infinite dimensions.

Characterized the semimartingale structure of the reflecting Ornstein-Uhlenbeck process.

Extended analysis to open sets defined by Borel functions.

Abstract

In this article we introduce the Gaussian Sobolev space $W^{1,2}(\mathscr O,\gamma)$, where $\mathscr O$ is an arbitrary open set of a separable Banach space $E$ endowed with a nondegenerate centered Gaussian measure $\gamma$. Moreover, we investigate the semimartingale structure of the infinite dimensional reflecting Ornstein-Uhlenbeck process for open sets of the form $\mathscr O=\{x\in E\, :\, G(x)<0\}$, where $ G$ is some Borel function on $E$.