Stochastic Heat Equations with Values in a Riemannian Manifold

I will talk about the existence of martingale solutions to the stochastic heat equation in a Riemannian manifold by using suitable Dirichlet form on the Riemannian path and loop space. Moreover, we present some characterizations of the lower bound of the Ricci curvature by functional inequalities of various associated Dirichlet forms.