Finding regular transport maps between measures is an important task in generative modelling and a useful tool to transfer functional inequalities. The most well-known result in this field is Caffarelli’s contraction theorem, which shows that the optimal transport map from a Gaussian to a uniformly log-concave measure is globally Lipschitz. Note that for our purposes optimality of the transport map does not play a role. This is why several works investigate other transport maps, such as those derived from diffusion processes, as introduced by Kim and Milman. Here, we establish a lower bound on the log-semiconcavity along the heat flow for a class of what we call asymptotically log-concave measures. We will see that this implies Lipschitz bounds for the heat flow map. It turns out that the very same log-semiconcavity bounds are also sufficient for proving exponential convergence of Sinkhorn's algorithm by following a recent work of Chiarini et al.

Our proof uses the control interpretation of the driving transport vector field inducing the transport map and a coupling strategy.

This talk is based on a joint work with Louis-Pierre Chaintion and Giovanni Conforti.