A variational approach to nonlinear and interacting diffusions

Marc Arnaudon (1) , Pierre del Moral (2, 3)

1 IMB - Institut de Mathématiques de Bordeaux
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
3 CQFD - Quality control and dynamic reliability

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.

Mots clés: Riemannian manifolds Mathematics; Nonlinear diffusions; Wasserstein distance; Contraction inequalities; Variational equations; Gradient flows; Riemannian manifold; Logarithmic norms; Mean field particle systems

Domaines: Probabilités [math.PR]

Marc Arnaudon, Pierre del Moral. A variational approach to nonlinear and interacting diffusions. 2019. ⟨hal-01950673v3⟩