Linearization of ergodic McKean SDEs and applications

TL;DR

This paper investigates the linearization of ergodic McKean SDEs and their PDEs, demonstrating exponential convergence of the nonlinear process to a linearized approximation, with applications to parameter estimation and particle systems.

Contribution

It introduces a linearized approach for ergodic McKean SDEs, providing convergence results and applications to estimation and particle system analysis.

Findings

Exponential convergence of the nonlinear process to the linearized process in entropy and Wasserstein distance.

Development of a linearized maximum likelihood estimator for the nonlinear process.

Analysis of the joint diffusive-mean field limit of interacting particle systems.

Abstract

In this article, we consider McKean stochastic differential equations, as well as their corresponding McKean-Vlasov partial differential equations, which admit a unique stationary state, and we study the linearized It\^o diffusion process that is obtained by replacing the law of the process in the convolution term with the unique invariant measure. We show that the law of the nonlinear McKean process converges to the law of this linearized process exponentially fast in time, both in relative entropy and in Wasserstein distance. We study the problem in both the whole space and the torus. We then show how we can employ the resulting linear (in the sense of McKean) Markov process to analyze properties of the original nonlinear and nonlocal dynamics that depend on their long-time behavior. In particular, we propose a linearized maximum likelihood estimator for the nonlinear process which is asymptotically unbiased, and we study the joint diffusive-mean field limit of the underlying interacting particle system.