Lipschitz continuity of diffusion transport maps from a control perspective
Abstract
Lipschitz transport maps between two measures are useful tools for transferring analytical properties, such as 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 the transfer of analytical properties does not depend on the optimality of the transport map. This is why several works have established Lipschitz bounds for other transport maps, such as those derived from diffusion processes, as introduced by Kim and Milman. Here, we use the control interpretation of the transport vector field inducing the transport map and a coupling strategy to obtain Lipschitz bounds for this map between asymptotically log-concave measures and their Lipschitz perturbations. This talk is based on a joint work with Giovanni Conforti.
