A SECOND ORDER ANALYSIS OF MCKEAN-VLASOV SEMIGROUPS

• 成果类型: Article
• 署名作者: Arnaudon, M.; Del Moral, P.
• 署名单位: Universite de Bordeaux; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Centre National de la Recherche Scientifique (CNRS); Inria; Institut Polytechnique de Paris; Ecole Polytechnique
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/20-AAP1568
• 发表日期: 2020
• 页码: 2613-2664
• 关键词: STOCHASTIC DIFFERENTIAL-EQUATIONS; Wasserstein Distance; STABILITY; CONVERGENCE; systems; propagation; derivatives; DIFFUSIONS; formulas

摘要

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-Grobner 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.