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).
@cite_1: We show that a certain entropy-like function is convex, under an optimal transport problem that is adapted to Ricci flow. We use this to reprove the monotonicity of Perelman’s reduced volume.
@cite_9: We present various characterizations of uniform lower bounds for the Ricci curvature of a smooth Riemannian manifold M in terms of convexity properties of the entropy (considered as a function on the space of probability measures on M) as well as in terms of transportation inequalities for volume measures, heat kernels, and Brownian motions and in terms of gradient estimates for the heat semigroup. © 2004 Wiley Periodicals, Inc.
@cite_4: In this note, we describe a new link between Perelman's monotonicity formula for the reduced volume and ideas from optimal transport theory.
McCann and Topping @cite_9 (see Topping @cite_4 and Lott @cite_1 also) showed contraction in the Wasserstein metric for the heat equation under backwards Ricci flow on a compact manifold. More precisely, they showed that the following are equivalent: @math evolves under backwards super Ricci flow, i.e. @math .