  In this paper, we are interested in the time derivative of the Wasserstein
distance between the marginals of two Markov processes. As recalled in the
introduction, the Kantorovich duality leads to a natural candidate for this
derivative. Up to the sign, it is the sum of the integrals with respect to each
of the two marginals of the corresponding generator applied to the
corresponding Kantorovich potential. For pure jump processes with bounded
intensity of jumps, we prove that the evolution of the Wasserstein distance is
actually given by this candidate. In dimension one, we show that this remains
true for Piecewise Deterministic Markov Processes. We apply the formula to
estimate the exponential decrease rate of the Wasserstein distance between the
marginals of two birth and death processes with the same generator in terms of
the Wasserstein curvature.
