Evolution of Gaussian Concentration Bounds under Diffusions

Abstract: We consider the behavior of the Gaussian concentration bound (GCB) under stochastic time evolution. More precisely, we consider a Markovian diffusion process on $\mathbb{R}^d$ and start the process from an initial distribution $\mu$ that satisfies GCB. We then study the question whether GCB is preserved under the time-evolution, and if yes, how the constant behaves as a function of time. In particular, if for the constant we obtain a uniform bound, then we can also conclude properties of the stationary measure(s) of the diffusion process. This question, as well as the methodology developed in the paper allows to prove Gaussian concentration via semigroup interpolation method, for measures which are not available in explicit form. We provide examples of conservation of GCB, loss of GCB in finite time, and loss of GCB at infinity. 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 a simple class of non-Markovian diffusion processes with drift of Ornstein-Uhlenbeck type, and general bounded predictable variance.