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2 results on '"Kinetic Fokker-Planck"'

1. IMPROVING THE CONVERGENCE RATES FOR THE KINETIC FOKKER-PLANCK EQUATION BY OPTIMAL CONTROL.

Author

BREITEN, TOBIAS and KUNISCH, KARL K.

Subjects
*FOKKER-Planck equation, *STOCHASTIC differential equations, *RICCATI equation, *LANGEVIN equations, *STOCHASTIC orders, *BILINEAR forms
Abstract

The long time behavior and detailed convergence analysis of Langevin equations have received increased attention in recent years. Difficulties arise from a lack of coercivity, usually termed hypocoercivity, of the underlying kinetic Fokker-Planck operator, which is a consequence of the partially deterministic nature of a second order stochastic differential equation. In this manuscript, the effect of controlling the confinement potential without altering the original invariant measure is investigated. This leads to an abstract bilinear control system with an unbounded but infinite-time admissible control operator which, by means of an artificial diffusion approach, is shown to possess a unique solution. The compactness of the underlying semigroup is further used to define an infinite-horizon optimal control problem on an appropriately reduced state space. Under smallness assumptions on the initial data, feasibility of and existence of a solution to the optimal control problem are discussed. Numerical results based on a local approximation based on a shifted Riccati equation illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

Published
2023
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2. Equations de Fokker-Planck cinétiques : hypocoercivité et hypoellipticité

Author

Cao, Chuqi, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Université Paris sciences et lettres, and Stéphane Mischler

Subjects
Fokker-Planck cinétiques, Hypocoercivity, Hypocoercivité, Convergence vers l’équilibre, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Hypoellipticity, Hypoellipticité, Kinetic Fokker-Planck, Boltzmann linéaire, Linear Boltzmann, Convergence to the equilibrium
Abstract

This thesis mainly study the hypocoercivity and long time behaviour of kinetic equations. We first consider the kinetic Fokker-Planck equation with weak confinement force and a class of general force. We prove the existence and uniqueness of a positive normalized equilibrium (in the case of a general force) and establish some exponential rate or sub-geometric rate of convergence to the equilibrium (and the rate can be explicitly computed). Then we study convergence to equilibriumof the linear relaxation Boltzmann (also known as linear BGK) and the linear Boltzmann equations either on the torus or on the whole space with a confining potential. We present explicit convergence results in total variation or weighted total variation norms. The convergence rates are exponential when the equations are posed on the torus, or with a confining potential growing at least quadratically at infinity. Moreover, we give algebraic convergence rates when subquadratic potentials considered. We use a method known as Harris’s Theorem.; Cette thèse porte principalement sur l’hypocoercivité et le comportement à long terme d’équations cinétiques. Nous considérons d’abord l’équation cinétique de Fokker-Planck avec la force de confinement faible et une classe de force générale. Nous prouvons l’existence et l’unicité d’un équilibre normalisé positif (dans le cas d’une force générale) et établissons un certain taux exponentiel ou sous-géométrique de convergence vers l’équilibre (et le taux peut être explicitement calculé). Ensuite, nous étudions la convergence vers l’équilibre de la relaxation Boltzmann linéaire (également appelé BGK linéaire) et le équations de Boltzmann linéaire soit sur le tore ou sur tout l’espace avec un confinement potentiel.Nous présentons des résultats de convergence explicites au normes de variation total ou de variation totale pondérée.Les taux de convergence sont exponentiels lorsque les équations sont posées sur le tore ou avec un potentiel de confinement grandir au moins quadratiquement à l’infini. De plus, nous donnons taux de convergence algébrique lorsque les potentiels sousquadratiqué pris en considération. Nous utilisons le théorème de Harris.

Published
2019

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