ARNAUDON, Marc; DEL MORAL, Pierre

doi:10.1214/20-AAP1568

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ARNAUDON, Marc
Institut de Mathématiques de Bordeaux [IMB]

DEL MORAL, Pierre
Centre de Mathématiques Appliquées de l'Ecole polytechnique [CMAP]
Quality control and dynamic reliability [CQFD]

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The Annals of Applied Probability. 2020

Institute of Mathematical Statistics (IMS)

Résumé en anglais

We propose a second order differential calculus to analyze the regularity and the stability properties of the distribution semigroup associated with McKean-Vlasov diffusions. This methodology provides second order Taylor type expansions with remainder for both the evolution semigroup as well as the stochastic flow associated with this class of nonlinear diffusions. Bismut-Elworthy-Li formulae for the gradient and the Hessian of the integro-differential operators associated with these expansions are also presented. The article also provides explicit Dyson-Phillips expansions and a refined analysis of the norm of these integro-differential operators. Under some natural and easily verifiable regularity conditions we derive a series of exponential decays inequalities with respect to the time horizon. We illustrate the impact of these results with a second order extension of the Alekseev-Gröbner lemma to nonlinear measure valued semigroups and interacting diffusion flows. This second order perturbation analysis provides direct proofs of several uniform propagation of chaos properties w.r.t. the time parameter, including bias, fluctuation error estimate as well as exponential concentration inequalities.< Réduire

Mots clés en anglais

Taylor expansions

Gradient flows

Logarithmic norms

Wasserstein distance

Contraction inequalities

Nonlinear diffusions

Mean field particle systems

Variational equations

Bismut-Elworthy-Li formulae

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A second order analysis of McKean-Vlasov semigroups

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