A coupling of Brownian motions in the ${\mathcal{L}}_0$-geometry

Abstract

Under a complete Ricci flow, we construct a coupling of two Brownian motion such that their ${\mathcal{L}}_0$-distance is a supermartingale. This recovers a result of Lott [J. Lott, Optimal transport and Perelman's reduced volume, Calc. Var. Partial Differential Equations 36 (2009), no. 1, 49--84.] on the monotonicity of ${\mathcal{L}}_0$-distance between heat distributions.