Hypercontractivity for perturbed diffusion semigroups

Abstract

Abstract. µ being a nonnegative measure satisfying some Log-Sobolev inequality, we give conditions on F for the Boltzmann measure ν = e −2F µ to also satisfy some Log-Sobolev inequality. This paper improves and completes the final section in [6]. A general sufficient condition and a general necessary condition are given and examples are explicitly studied.Résumé. µétant une mesure positive satisfaisant une inégalité de Sobolev logarithmique, nous donnons des conditions sur F pour que la mesure de Boltzmann ν = e −2F µ satisfasseégalement une telle inégalité (améliorant et complétant ainsi la dernière partie de [6]). Les conditions obtenues sont illustrées par des exemples.

Long time behavior of Markov processes

“…(2) If lim inf ∞ (β|∇V | 2 − ∆V ) > 0, for some β < 1, then there exists some Foster-Lyapunov function. This extends some previous results by Shigeo Kusuoka and Daniel W. Stroock (see [13]) and similar results obtained by the ground state transformation (that transforms the Fokker-Planck operator into a Schrödinger one, whose spectral theory is better known) yielding the so called Witten Laplacian of Bernard Helffer and Francis Nier [34]. (3) In one dimension one can replace the bounded perturbation by a super-linear perturbation (see [23]).…”

“…In the log-Sobolev case this was first proved by Kusuoka and Stroock ( [37] also see [13]). Similarly, drift conditions like < x, ∇V (x) >≥ α |x| β − c for all x, some β > 1 and some α > 0 yield a F -Sobolev inequality for F (u) = ln…”

Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry

“…In the general case these assumptions are denoted by (H.F) in [21]. Here we have chosen the usual definition…”

“…Section 2 contains the required elements on Orlicz spaces. Section 3 presents a sufficient condition on the Young function τ for Q α t (a slightly modified P α t ) to map continuously L 2 into L τ , for a fixed t. This condition relies on the probabilistic representation of P α t (Girsanov transformation) and on martingale methods inspired by [35,21]. Unfortunately the method cannot reach the contraction property (only boundedness, simply called τ -Orlicz hyperboundedness) and does not easily yield explicit bounds.…”

Logarithmic Sobolev inequalities and spectral gaps

“…See [1] for an insightful recent survey, and of particular relevance here, recent work of Cattiaux [3].…”