Stochastic Heat Equations with Values in a Manifold via Dirichlet Forms

TL;DR

This paper establishes the existence of martingale solutions to stochastic heat equations valued in Riemannian manifolds using Dirichlet forms, and explores related inequalities and curvature bounds.

Contribution

It introduces a novel approach to solving stochastic heat equations on manifolds via Dirichlet forms and derives key inequalities and curvature characterizations.

Findings

Existence of martingale solutions with invariant Wiener measure

Log-Sobolev inequality established for the Dirichlet form

Characterizations of Ricci curvature bounds related to the equation

Abstract

In this paper, we prove the existence of martingale solutions to the stochastic heat equation taking values in a Riemannian manifold, which admits Wiener (Brownian bridge) measure on the Riemannian path (loop) space as an invariant measure using a suitable Dirichlet form. Using the Andersson-Driver approximation, we heuristically derive a form of the equation solved by the process given by the Dirichlet form. Moreover, we establish the log-Sobolev inequality for the Dirichlet form in the path space. In addition, some characterizations for the lower bounds of the Ricci curvature are presented related to the stochastic heat equation.