Stochastic Heat Equations for infinite strings with Values in a Manifold

TL;DR

This paper constructs Markov processes for stochastic heat equations on infinite strings valued in Riemannian manifolds, extending finite volume results and analyzing ergodic properties based on curvature conditions.

Contribution

It extends the construction of Markov processes for stochastic heat equations to infinite strings with manifold values, and establishes functional inequalities and ergodic behavior.

Findings

Exponential ergodicity when Ricci curvature is positive

Non-ergodicity when sectional curvature is negative

Extension of finite volume results to infinite volume case

Abstract

In the paper, we construct conservative Markov processes corresponding to the martingale solutions to the stochastic heat equation on $\mathbb{R}^+$ or $\mathbb{R}$ with values in a general Riemannian maifold, which is only assumed to be complete and stochastic complete. This work is an extension of the previous paper \cite{RWZZ17} on finite volume case. Moveover, we also obtain some functional inequalities associated to these Markov processes. This implies that on infinite volume case, the exponential ergodicity of the solution if the Ricci curvature is strictly positive and the non-ergodicity of the process if the sectional curvature is negative.