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A second order analysis of McKean-Vlasov semigroups | | M Arnaudon
; P Del Moral
; | | Date: |
12 Jun 2019 | | 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-Gr{"o}bner 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. | | Source: | arXiv, 1906.5140 | | Services: | Forum | Review | PDF | Favorites | |
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