A coupling of Brownian motions in the $\mathcal{L}_0$-geometry

TL;DR

This paper constructs a coupling of two Brownian motions under Ricci flow ensuring their $ abla_0$-distance is a supermartingale, thereby confirming the monotonicity of $ abla_0$-distance between heat distributions as previously shown by Lott.

Contribution

It introduces a novel coupling method for Brownian motions in $ abla_0$-geometry under Ricci flow, extending the understanding of heat distribution behavior.

Findings

$ abla_0$-distance is a supermartingale under the coupling.

Monotonicity of $ abla_0$-distance between heat distributions is established.

Provides a probabilistic proof of Lott's result.

Abstract

Under a complete Ricci flow, we construct a coupling of two Brownian motion such that their $\mathcal{L}_0$-distance is a supermartingale. This recovers a result of Lott [J. Lott, Optimal transport and Perelman's reduced volume, Calc. Var. Partial Differential Equations 36 (2009), no. 1, 49--84.] on the monotonicity of $\mathcal{L}_0$-distance between heat distributions.