Relaxation to equilibrium for kinetic Fokker-Planck equation

Change log
Piazzoli, Davide 

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.

Mouhot, Clément
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
Master of Science (MSc)
Awarding Institution
University of Cambridge