The radial part of Brownian motion with respect to $\mathcal{L}$-distance under Ricci flow

TL;DR

This paper develops an Itô formula for the al-distance of Brownian motion under Ricci flow and uses it to construct couplings, providing new proofs of existing results.

Contribution

It introduces an Itô formula for al-distance under Ricci flow and constructs couplings, advancing stochastic analysis in evolving geometric settings.

Findings

Established Itô formula for al-distance under Ricci flow.

Constructed a coupling by parallel displacement.

Provided new proofs of Topping's results.

Abstract

Let $\{g_t\}_{t\in [0,T)}$ be a family of complete time-depending Riemannian matrics on a manifold which evolves under backwards Ricci flow. The It\^{o} formula is established for the $\mathcal{L}$-distance of the $g_t$-Brownian motion to a fixed reference point ($\mathcal{L}$-base). Furthermore, as an application, we construct a coupling by parallel displacement which yields a new proof of some results of Topping.