A Class of infinite dimensional stochastic Processes with unbounded Diffusion

TL;DR

This paper explores a class of infinite-dimensional stochastic processes with unbounded diffusion operators, analyzing their properties on Wiener spaces and under measure changes, with implications for stochastic analysis on manifolds.

Contribution

It introduces a framework for studying Dirichlet forms with unbounded diffusion operators on Wiener spaces, extending results to Riemannian manifolds and showing measure invariance of quasi-regularity.

Findings

Quasi-regularity is preserved under certain measure changes.

Derivative and divergence operators are closable with inverses.

Results are transferred from classical Wiener space to manifold-based Wiener space.

Abstract

The paper studies Dirichlet forms on the classical Wiener space and the Wiener space over non-compact complete Riemannian manifolds. The diffusion operator is almost everywhere an unbounded operator on the Cameron--Martin space. In particular, it is shown that under a class of changes of the reference measure, quasi-regularity of the form is preserved. We also show that under these changes of the reference measure, derivative and divergence are closable with certain closable inverses. We first treat the case of the classical Wiener space and then we transfer the results to the Wiener space over a Riemannian manifold.