Convergence of Brownian Motions on Metric Measure Spaces Under Riemannian Curvature-Dimension Conditions

TL;DR

This paper establishes a link between the geometric convergence of metric measure spaces satisfying RCD conditions and the weak convergence of associated Brownian motions, enhancing understanding of stochastic processes in geometric analysis.

Contribution

It proves that pointed measured Gromov convergence of spaces under RCD conditions implies the weak convergence of Brownian motions, connecting geometric and probabilistic convergence.

Findings

Pointed measured Gromov convergence implies weak convergence of Brownian motions.

Under certain conditions, the two types of convergence are equivalent.

The results deepen the understanding of stochastic processes on converging geometric spaces.

Abstract

We show that the pointed measured Gromov convergence of the underlying spaces implies (or under some condition, is equivalent to) the weak convergence of Brownian motions under Riemannian Curvature-Dimension (RCD) conditions.