ARNAUDON, Marc; DEL MORAL, Pierre

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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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Document de travail - Pré-publication

Résumé en anglais

The article presents a novel variational calculus to analyze the stability and the propagation of chaos properties of nonlinear and interacting diffusions. This differential methodology combines gradient flow estimates with backward stochastic interpolations, Lyapunov linearization techniques as well as spectral theory. This framework applies to a large class of stochastic models including non homogeneous diffusions, as well as stochastic processes evolving on differentiable manifolds, such as constraint-type embedded manifolds on Euclidian spaces and manifolds equipped with some Riemannian metric. We derive uniform as well as almost sure exponential contraction inequalities at the level of the nonlinear diffusion flow, yielding what seems to be the first result of this type for this class of models. Uniform propagation of chaos properties w.r.t. the time parameter are also provided. Illustrations are provided in the context of a class of gradient flow diffusions arising in fluid mechanics and granular media literature. The extended versions of these nonlinear Langevin-type diffusions on Riemannian manifolds are also discussed.< Réduire

Mots clés en anglais

Riemannian manifolds Mathematics

Nonlinear diffusions

Wasserstein distance

Contraction inequalities

Variational equations

Gradient flows

Riemannian manifold

Logarithmic norms

Mean field particle systems

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A variational approach to nonlinear and interacting diffusions

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Résumé en anglais

Mots clés en anglais

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