Evolution of Gaussian Concentration Bounds under Diffusions

J.-R. Chazottes, P. Collet, F. Redig

2021, v.27, Issue 5, 707-754

ABSTRACT

We consider the behavior of the Gaussian concentration bound \linebreak (GCB)
under stochastic time evolution.
More precisely, we consider a Markovian diffusion process on $\R^d$ and start
the process from an initial distribution $\mu$ that satisfies GCB. We then s
tudy the question
whether GCB is preserved under the time-evolution, and if yes, how the consta
nt 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.

Keywords: Markov diffusions, Ornstein-Uhlenbeck process, nonlinear semigroup, coupling, Bakry-Emery criterion, non-reversible diffusions, diffusions coming down from infinity, Ginzburg-Landau diffusions, non-Markovian diffusions, Lorenz attractor with noise, Burkholder inequality

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