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Relaxation to equilibrium for kinetic Fokker-Planck equation


Type

Thesis

Change log

Authors

Piazzoli, Davide 

Abstract

We want to study long-time behaviour of solutions ft of kinetic Fokker-Planck equation in Rd, namely their convergence towards equilibrium f in the form [ \textrm{d}(f_t,f_\infty)\leq C_1 e^{-C_2 t}\textrm{d}(f_0,\mu) ] for appropriate distances d and constants C1≥1, C2>0.

In Section 1 we provide an introduction and motivation for the equation, together with the setting of {Villani, Hypocoercivity} which will be useful in Section 2.

In Section 2 we will review the monograph {Villani, Hypocoercivity}, where such convergence is proved, for h=f/μ, in H1(μ) and Hμ+Iμ, that is, the sum of relative entropy and Fisher information. Here results are stated in terms of general operators t+AA+B=0, and commutation conditions on A and B are to be imposed.

In Section 3 we shall take into consideration the work by Monmarch'{e} {Monmarche, Generalized Γ calculus} in which such convergence is established by rephrasing some concepts in term of Γ-calculus: with respect to {Villani, Hypocoercivity} there is no need for regularization along the semigroup since the functional taken into account is a modified H+I that at initial time only takes entropy into account, and the argument turns out to be shorter. Also, the convergence rate is eCt(1−et)2 instead of C1eC2t. However it turns out, as in {Villani, Hypocoercivity}, that for this case it is strictly needed to have a pointwise bound on D2U, where U is the confinement potential. A drawback of this method with respect to {Villani, Hypocoercivity} is that, in a more general setting than kinetic Fokker-Planck equation, stronger commutation assumptions are required, which imply that the diffusion matrix is basically required to be constant. On this work a specific analysis was carried out, simplifying the proof for our Fokker-Planck case and finding explicit and improved expressions for convergence constants.

The same author in {Monmarche, chaos kinetic particles}, which is the subject of Section 4, addresses a Vlasov-Fokker-Planck equation with a potential that generalizes U and the related particle system. Chaos propagation in W2, the 2-Wasserstein distance, is proved, namely W2(ft(1,N),ft)≤CNϵ. This leads to both Wasserstein and L1 hypocoercivity, however dependence of the right hand side from the initial data is not linear as wished.

Description

Date

2017-10-01

Advisors

Mouhot, Clément

Keywords

Villani, hypocoercivity, kinetic, PDE, Wasserstein, entropy, Monmarche, Fokker, Planck, Fokker-Planck, Logarithmic Sobolev inequality, Gamma calculus, carré du champ, commutation, Mouhot, Bolley, particle system, Bakry-Emery

Qualification

Master of Science (MSc)

Awarding Institution

University of Cambridge