HAL CCSD
Construction of Boltzmann and McKean Vlasov type flows (the sewing lemma approach)
Alfonsi, Aurélien
Bally, Vlad
Mathematical Risk Handling (MATHRISK) ; Université Paris-Est Marne-la-Vallée (UPEM)-École des Ponts ParisTech (ENPC)-Inria de Paris ; Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)
Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS) ; École des Ponts ParisTech (ENPC)
Laboratoire Analyse et Mathématiques Appliquées (LAMA) ; Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel
AA acknowledges the support of the “chaire Risques financiers”, Fondation du Risque
hal-03241604
https://enpc.hal.science/hal-03241604
https://enpc.hal.science/hal-03241604
2021
ARXIV: 2105.12677
info:eu-repo/semantics/altIdentifier/arxiv/2105.12677
en
Sewing lemma
Boltzmann equation
Enskog equation
McKean-Vlasov equation
Interacting particle system
2010 Mathematics Subject Classification.35Q20 35Q83 76P05 60H20
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]
info:eu-repo/semantics/preprint
Preprints, Working Papers, ...
We are concerned with a mixture of Boltzmann and McKean-Vlasov type equations, this means (in probabilistic terms) equations with coefficients depending on the law of the solution itself,and driven by a Poisson point measure with the intensity depending also on the law of the solution. Both the analytical Boltzmann equation and the probabilistic interpretation initiated by Tanaka (1978) have intensively been discussed in the literature for specific models related to the behavior of gas molecules. In this paper, we consider general abstract coefficients that may include mean field effects and then we discuss the link with specific models as well. In contrast with the usual approach in which integral equations are used in order to state the problem, we employ here a new formulation of the problem in terms of flows of endomorphisms on the space of probability measure endowed with the Wasserstein distance. This point of view already appeared in the framework of rough differential equations. Our results concern existence and uniqueness of the solution, in the formulation of flows, but we also prove that the "flow solution" is a solution of the classical integral weak equation and admits a probabilistic interpretation. Moreover, we obtain stability results and regularity with respect to the time for such solutions. Finally we prove the convergence of empirical measures based on particle systems to the solution of our problem, and we obtain the rate of convergence. We discuss as examples the homogeneous and the inhomogeneous Boltzmann (Enskog) equation with hard potentials.
2021-05-28