Large deviations for Brownian motion in evolving Riemannian manifolds

2020/04/01 by Rik Versendaal, Versendaal, Rik
Mathematics · #58J65 #60F10 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2004.00358

openalex publication_date 2020/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove large deviations for g(t)-Brownian motion in a complete, evolving Riemannian manifold M with respect to a collection \g(t)\_{t∈ [0,1]} of Riemannian metrics, smoothly depending on t. We show how the large deviations are obtained from the large deviations of the (time-dependent) horizontal lift of g(t)-Brownian motion to the frame bundle FM over M. The latter is proved by embedding the frame bundle into some Euclidean space and applying Freidlin-Wentzell theory for diffusions with time-dependent coefficients, where the coefficients are jointly Lipschitz in space and time.

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