  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.
