Evolution of Gaussian concentration bounds under diffusions
Résumé
We 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 available
in 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.
Mots clés
non Markovian diffusion
nonlinear semigroup
Gaussian concentration bound
diffusion processes
Ornstein-Uhlenbeck process
coupling
diffusion coming from infinity
Key-words: Markov diffusions
nonlin- ear semigroup
Bakry-Emery criterion
non-reversible dif- fusions
diffusions coming down from infinity
Ginzburg-Landau diffu- sions
non-Markovian diffusions
Lorenz attractor with noise
Burkholder inequality *
Domaines
Probabilités [math.PR]Origine | Fichiers produits par l'(les) auteur(s) |
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