Coupling of Brownian motions and Perelman’s L-functional

We show that on a manifold whose Riemannian metric evolves under backwards Ricci flow two Brownian motions can be coupled in such a way that their normalized L-distance is a supermartingale. As a corollary, we obtain the monotonicity of the transportation cost between two solutions of the heat equation in the case that the cost function is the composition of a concave non-decreasing function and the normalized L-distance. In particular, it provides a new proof of a recent result of Topping [P. Topping, L-optimal transportation for Ricci flow, J. Reine Angew. Math. 636 (2009) 93–122]. © 2011 Elsevier Inc. All rights reserved.