The Kinetic Fokker-planck Equation With Mean Field Interaction

Abstract

We study the long time behaviour of the kinetic Fokker-Planck equation with mean field interaction, whose limit is often called Vlasov-Fkker-Planck equation. We prove a uniform (in the number of particles) exponential convergence to equilibrium for the solutions in the weighted Sobolev space H 1 (µ) with a rate of convergence which is explicitly computable and independent of the number of particles. The originality of the proof relies on functional inequalities and hypocoercivity with Lyapunov type conditions, usually not suitable to provide adimensional results.

Entropic turnpike estimates for the kinetic Schrödinger problem

“…[4,21,29,47] and references therein, or a probabilistic approach, see e.g. [23,26], as well as on the new developments around the long-time behaviour of Schrödinger bridges, in order to gain some understanding on controlled versions of the kinetic Fokker-Planck equation. Leaving a more accurate comparison between our results and the existing literature to the text below, let us first present a very concise summary of our contributions and explain how this article is structured.…”

“…Although exponential L 2 estimates are known to hold under considerably weaker assumptions (see e.g. [29]), and entropic estimates assuming a bounded and positive Hessian have been known for more than a decade [47], it is only recently [26] that entropic estimates have been obtained beyond the bounded Hessian case. In light of this, the question of how to improve our results is quite interesting and deserves to be further investigated.…”

“…for some well chosen positive definite symmetric matrix M ∈ R 2d×2d . This is a common idea in the kinetic setting and it is exploited for example in [4,26,36].…”

Hypocoercivity and global hypoellipticity for the kinetic Fokker-Planck equation in $H^k$ spaces

“…The exponents in our regularity estimates are optimal in short time. Moreover, as in our recent work [30], the general results here can be applied in the mean-field setting to get estimates independent of the dimension; in particular, an application to the Curie-Weiss model is presented.…”

“…Moreover, hypocoercivity can be measured in many different ways, such as entropy, ϕentropies, total variation, Wasserstein distance, L 2 norms, H k norms, or more general weighted Sobolev norms, see for instance [5], [9], [3], [41], [16], [21], [7], [24](see also related work [22] and [23]), [25] and the references therein. We also call attentions to [39], [18], [10], [11], [12], [15], [14] and [30] for some recent results concerning hypocoercivity with various kinds of potentials. In particular, we highlight [8] and [37] for some results concerning singular potentials.…”