Katharina Eichinger - Properties of diffusion transport maps via creation of log-semiconcavity... скачать в хорошем качестве

Katharina Eichinger - Properties of diffusion transport maps via creation of log-semiconcavity...

This lecture was part of the Workshop on "Probabilistic Mass Transport - from Schrödinger to Stochastic Analysis" held at the ESI February 9 - 13, 2026. 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. I will also comment on its implication for stability of these maps. Based on a joint work with Louis-Pierre Chaintron and Giovanni Conforti