Non-explosion of diffusion processes on manifolds with time-dependent metric

TL;DR

This paper proves that diffusion processes, specifically Brownian motion, do not explode in finite time on manifolds with a time-dependent metric evolving under backwards Ricci flow, extending previous non-explosion results.

Contribution

It establishes non-explosion of Brownian motion under backwards Ricci flow and generalizes an Ito formula for distance functions on manifolds.

Findings

Brownian motion cannot explode under backwards Ricci flow

Derived an Ito formula for distance from a fixed point

Removed the non-explosion assumption in a key contraction result

Abstract

We study the problem of non-explosion of diffusion processes on a manifold with time-dependent Riemannian metric. In particular we obtain that Brownian motion cannot explode in finite time if the metric evolves under backwards Ricci flow. Our result makes it possible to remove the assumption of non-explosion in the pathwise contraction result established by Arnaudon, Coulibaly and Thalmaier (arXiv:0904.2762, to appear in Sem. Prob.). As an important tool which is of independent interest we derive an Ito formula for the distance from a fixed reference point, generalising a result of Kendall (Ann. Prob. 15 (1987), 1491--1500).