Temporal and spatial discretizations of Brownian motion on Riemannian manifolds

by Simon Schwarz
Doctoral thesis
Date of Examination:2025-05-26
Date of issue:2025-09-02
Advisor:Prof. Dr. Anja Sturm
Referee:Prof. Dr. Anja Sturm
Referee:Prof. Dr. Max Wardetzky
Persistent Address: http://dx.doi.org/10.53846/goediss-11480

 

 

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Abstract

English

This thesis studies temporal and spatial discretizations of Brownian motion and diffusion processes on Riemannian manifolds. Geodesic random walks serve as temporal discretizations of diffusion processes and converge to the respective Brownian motion, but evaluating geodesics can be computationally costly. Motivated by the need for efficient and simple, yet convergent approximations of diffusion processes on manifolds, we introduce retraction-based random walks and prove convergence to Brownian motion under a second-order condition. We provide concrete examples of second-order retractions and also extend this framework to sub-Riemannian manifolds. For spatial discretizations, we define discrete Brownian motion via the cotan Laplacian, establish a functional central limit theorem, and investigate short-time asymptotics of the discrete heat kernel, identifying distinct continuous and discrete regimes. Finally, we prove convergence of the heat method for geodesic distance computation using homogenization techniques.
Keywords: Brownian motion; Riemannian manifolds; Heat method; Retractions; Geodesic random walk; Short-time asymptotics; Discrete differential geometry
 
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