Evolution of Gaussian concentration bounds under diffusions

International audienceWe study the behavior of the Gaussian concentration bound (GCB) under stochastic time evolution.More precisely, in the context of Markovian diffusion processes on $\mathbb{R}^d$ we prove in various settings that if we start the process from an initial probability measure satisfying GCB, then at later times GCB holds, and estimates for the constant are provided. Under additional conditions, we show that GCB holds for the unique invariant measure.This gives a semigroup interpolation method to prove Gaussian concentration for measures which are not availablein explicit form.We also consider diffusions ``coming down from infinity'' for which we show that, from any starting measure,at positive times, GCB holds.Finally we consider non-Markovian difussion processes with drift of Ornstein Uhlenbeck type, and general bounded predictable variance