Analysis on Path Spaces over Riemmannian Manifolds with Boundary

TL;DR

This paper develops a damped gradient operator for reflecting Brownian motion on compact Riemannian manifolds with boundary, establishing a log-Sobolev inequality on the path space using Hsu's multiplicative functional.

Contribution

It introduces a new damped gradient operator based on Hsu's multiplicative functional, connecting it to quasi-invariant flows and deriving a log-Sobolev inequality.

Findings

Defined a natural damped gradient operator for reflecting Brownian motion.

Established an integration by parts formula linking the operator to quasi-invariant flows.

Proved a standard log-Sobolev inequality for the Dirichlet form on the path space.

Abstract

By using Hsu's multiplicative functional for the Neumann heat equation, a natural damped gradient operator is defined for the reflecting Brownian motion on compact manifolds with boundary. This operator is linked to quasi-invariant flows in terms of a integration by parts formula, which leads to the standard log-Sobolev inequality for the associated Dirichlet form on the path space.