# A variational approach to nonlinear and interacting diffusions

## TL;DR

This paper introduces a new variational calculus framework combining multiple techniques to analyze stability and chaos propagation in nonlinear, interacting diffusions across diverse stochastic models and manifolds.

## Contribution

It develops a unified variational approach integrating gradient flows, stochastic interpolations, and spectral theory for nonlinear diffusions, including on manifolds, with new exponential contraction results.

## Key findings

- First exponential contraction inequalities for this class of models.
- Uniform propagation of chaos over time.
- Applications to fluid mechanics and granular media.

## Abstract

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.