# Stochastic Heat Equations with Values in a Riemannian Manifold

## TL;DR

This paper establishes the existence of martingale solutions to stochastic heat equations on Riemannian manifolds and explores how Ricci curvature bounds relate to functional inequalities of Dirichlet forms.

## Contribution

It introduces a method using Dirichlet forms on path and loop spaces to prove solution existence and characterizes Ricci curvature bounds via functional inequalities.

## Key findings

- Existence of martingale solutions to SHE on Riemannian manifolds
- Characterization of Ricci curvature lower bounds through Dirichlet form inequalities
- Connection between geometric curvature bounds and stochastic analysis

## Abstract

The main result of this note is the existence of martingale solutions to the stochastic heat equation (SHE) in a Riemannian manifold by using suitable Dirichlet forms on the corresponding path/loop space. Moreover, we present some characterizations of the lower bound of the Ricci curvature by functional inequalities of various associated Dirichlet forms.