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Constructive Krein-Rutman result for Kinetic Fokker-Planck equations in a domain

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arxiv 2407.10530 v3 pith:Z33DXTFL submitted 2024-07-15 math.AP

Kleber Carrapatoso , Pierre Gabriel , Richard Medina , Stephane Mischler This is my paper · ORCID
classification math.AP
keywords constructivedomainequationestablishfokker-planckfundamentalkinetickrein-rutman
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We consider a general Kinetic Fokker-Planck (KFP) equation in a domain with Maxwell reflection condition on the boundary, not necessarily with conservation of mass. We establish the wellposedness in many spaces including Radon measures spaces, and in particular the existence and uniqueness of fundamental solutions. We also establish a Krein-Rutman theorem with constructive rate of convergence in an abstract setting that we use for proving that the solutions to the KFP equation converge toward the conveniently normalized first eigenfunction. Both results use the ultracontractivity of the associated semigroup in a fundamental way.

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  1. Sharp kinetic trace theory

    math.AP 2026-07 accept novelty 8.0 of 10

    Natural kinetic traces hold on half-spaces unrestricted, fail for unrestricted Gaussian when p<2 on every bounded C^{1,1} domain, and for bounded velocities are sharp exactly at boundary regularity α_p=1/(p+1).

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