Lipschitz estimates on the JKO scheme for the Fokker-Plack equation on bounded convex domains
Résumé
Given a semi-convex potential V on a convex and bounded domain Ω, we consider the Jordan-Kinderlehrer-Otto scheme for the Fokker-Planck equation with potential V, which defines, for fixed time step τ > 0, a sequence of densities ρ k ∈ P(Ω). Supposing that V is α-convex, i.e. D 2 V ≥ αI, we prove that the Lipschitz constant of log ρ + V satisfies the following inequality: Lip(log(ρ k+1) + V)(1 + ατ) ≤ Lip(log(ρ k) + V). This provides exponential decay if α > 0, Lipschitz bounds on bounded intervals of time, which is coherent with the results on the continuous-time equation, and extends a previous analysis by Lee in the periodic case.
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Commentaire : Ce pdf est la version preprint de l'article (version soumise à l'éditeur, avant peer-reviewing)
Commentaire : Ce pdf est la version preprint de l'article (version soumise à l'éditeur, avant peer-reviewing)
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