Stochastic Heat Equations for infinite strings with Values in a Manifold

2018/12/05 by Xin Chen, Chen, Xin, Bo Wu +5
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.1812.01942

openalex publication_date 2018/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In the paper, we construct conservative Markov processes corresponding to the martingale solutions to the stochastic heat equation on ℝ⁺ or 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 \citeRWZZ17 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.

Citations

• The motion of a random string

Related

• Stochastic Heat Equations with Values in a Riemannian Manifold
• Stochastic Heat Equations with Values in a Manifold via Dirichlet Forms
• Nonlinear characterizations of stochastic completeness
• Stochastic Heat Equations defined by Fractal Laplacians on Cantor-like Sets
• Moment estimates for the stochastic heat equation on Cartan-Hadamard manifolds