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

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 et?al. (arXiv:0904.2762, to appear in Sém. Prob.). As an important tool which is of independent interest we derive an It? formula for the distance from a fixed reference point, generalising a result of Kendall (Ann. Prob. 15:1491?C1500, 1987).