Convergence of Brownian motions on RCD*(K,N) spaces

TL;DR

This paper establishes that for a sequence of RCD*(K,N) spaces with bounded diameter, Sturm's D-convergence is equivalent to the weak convergence of Brownian motion laws, linking geometric and probabilistic convergence.

Contribution

It proves the equivalence between Sturm's D-convergence and weak convergence of Brownian motions on RCD*(K,N) spaces with bounded diameter.

Findings

Sturm's D-convergence matches weak Brownian motion law convergence.

The result applies to sequences of RCD*(K,N) spaces with bounded diameter.

Provides a bridge between geometric and probabilistic convergence notions.

Abstract

Suppose that a sequence of metric measure spaces X_n=(X_n, d_n, m_n) satisfies RCD*(K,N) with Diam(X_n) <D and m_n(X_n)=1. Then Sturm's D-convergence of X_n is equivalent to the weak convergence of the laws of Brownian motions on X_n.