Stochastic Heat Equations with Values in a Manifold via Dirichlet Forms

2017/11/27 by Michael Röckner, Röckner, Michael, Bo Wu +5
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Probability (math.PR) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1711.09570

openalex publication_date 2017/11/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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