A second order analysis of McKean-Vlasov semigroups
DEL MORAL, Pierre
Centre de Mathématiques Appliquées - Ecole Polytechnique [CMAP]
Quality control and dynamic reliability [CQFD]
Centre de Mathématiques Appliquées - Ecole Polytechnique [CMAP]
Quality control and dynamic reliability [CQFD]
DEL MORAL, Pierre
Centre de Mathématiques Appliquées - Ecole Polytechnique [CMAP]
Quality control and dynamic reliability [CQFD]
< Leer menos
Centre de Mathématiques Appliquées - Ecole Polytechnique [CMAP]
Quality control and dynamic reliability [CQFD]
Idioma
en
Article de revue
Este ítem está publicado en
The Annals of Applied Probability. 2020
Institute of Mathematical Statistics (IMS)
Resumen en inglés
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 ...Leer más >
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.< Leer menos
Palabras clave en inglés
Taylor expansions
Gradient flows
Logarithmic norms
Wasserstein distance
Contraction inequalities
Nonlinear diffusions
Mean field particle systems
Variational equations
Bismut-Elworthy-Li formulae
Orígen
Importado de HalCentros de investigación