Convergence of Brownian motions on RCD(K,infty) spaces

TL;DR

This paper establishes that in RCD(K,∞) metric measure spaces, the convergence of spaces in the measured Gromov sense is equivalent to the weak convergence of their associated Brownian motion laws, linking geometric and probabilistic convergence.

Contribution

It proves the equivalence between measured Gromov convergence and weak convergence of Brownian motion laws in RCD(K,∞) spaces, connecting geometric and stochastic analysis.

Findings

Measured Gromov convergence implies weak convergence of Brownian motions.

Weak convergence of Brownian motions characterizes measured Gromov convergence.

Results apply to spaces with total measure one.

Abstract

Suppose that metric measure spaces X_n=(X_n, d_n, m_n) satisfy RCD(K,infty) conditions with m_n(X_n)=1. Then the measured Gromov convergence (introduced by Gigili-Mondino-Savare '13) of X_n is equivalent to the weak convergence of the laws of Brownian motions on X_n with initial distributions m_n.