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_0: 
@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.
Whenever @math and @math are two non-negative unit-mass solutions of the heat equation [ u t = 1 2 g(t) u - ( 1 2 g t ) u ] (the term @math comes from the change in time of the volume element), the function @math is non-increasing. Here [ W_2(t, , ) := ( M M d_ g(t) (x,y)^2 (dx, dy) )^ 1 2 ] is the @math -Wasserstein distance of two probability measures @math and @math on @math . (The infimum is over all probability measures @math on @math whose marginals are @math and @math .) It means that backwards super Ricci flow is characterised by the contractivity property for solutions of the heat equation. Moreover, in recent work by Topping @cite_4 and Lott @cite_1 (see Brendle @cite_0 also) the heat equation and the theory of optimal transport are efficiently used to derive several monotonicity results including a new proof for the monotonicity of Perelman's reduced volume. These facts indicate that it would be effective for deeper understanding of Ricci flow to study the heat equation in conjunction with backwards Ricci flow and the theory of optimal transport.