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.
Cite this article
Michael Röckner, Bo Wu, Rongchan Zhu, Xiangchan Zhu, Stochastic heat equations with values in a Riemannian manifold. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 29 (2018), no. 1, pp. 205–213DOI 10.4171/RLM/801