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December 2020 A second order analysis of McKean–Vlasov semigroups
M. Arnaudon, P. Del Moral
Ann. Appl. Probab. 30(6): 2613-2664 (December 2020). DOI: 10.1214/20-AAP1568

Abstract

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.

Citation

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M. Arnaudon. P. Del Moral. "A second order analysis of McKean–Vlasov semigroups." Ann. Appl. Probab. 30 (6) 2613 - 2664, December 2020. https://doi.org/10.1214/20-AAP1568

Information

Received: 1 June 2019; Revised: 1 December 2019; Published: December 2020
First available in Project Euclid: 14 December 2020

Digital Object Identifier: 10.1214/20-AAP1568

Subjects:
Primary: 47J20, 58J65, 65C35, 82C80

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.30 • No. 6 • December 2020
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