We consider the behavior of the Gaussian concentration bound (GCB) under stochastic time evolution. More precisely, we consider a Markovian diffusion process on ℝ^d and start the process from an initial distribution µ 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.

We consider equilibrium states (that is, shift-invariant Gibbs measures) on the configuration space SZd where d≥ 1 and S is a finite set. We prove that if an equilibrium state for a shift-invariant uniformly summable potential satisfies a Gaussian concentration bound, then it is unique. Equivalently, if there exist several equilibrium states for a potential, none of them can satisfy such a bound.

We consider Gibbs measures on the configuration space (Formula presented.), where mostly (Formula presented.) and S is a finite set. We start by a short review on concentration inequalities for Gibbs measures. In the Dobrushin uniqueness regime, we have a Gaussian concentration bound, whereas in the Ising model (and related models) at sufficiently low temperature, we control all moments and have a stretched-exponential concentration bound. We then give several applications of these inequalities whereby we obtain various new results. Amongst these applications, we get bounds on the speed of convergence of the empirical measure in the sense of Kantorovich distance, fluctuation bounds in the Shannon–McMillan–Breiman theorem, fluctuation bounds for the first occurrence of a pattern, as well as almost-sure central limit theorems.