Common noise pullback attractors for stochastic dynamical systems

TL;DR

This paper studies stochastic differential equations driven by intrinsic and common noise, proving the existence of unique pullback attractors that are smooth densities depending only on the past of the common noise, with implications for particle systems.

Contribution

It establishes the almost sure existence and uniqueness of common noise pullback attractors for SDEs, linking them to stochastic Fokker-Planck equations and particle system distributions.

Findings

Existence and uniqueness of common noise pullback attractors.

Attractors are smooth probability densities.

Convergence of the stochastic Fokker-Planck equation to attractors.

Abstract

We consider SDEs driven by two different sources of additive noise, which we refer to as intrinsic and common. We establish almost sure existence and uniqueness of pullback attractors with respect to realisations of the common noise only. These common noise pullback attractors are smooth probability densities that depend only on (the past of) a common noise realisation and to which the pullback evolution of a corresponding stochastic Fokker-Planck equation converges. Common noise pullback attractors have a natural motivation in the context of particle systems with intrinsic and common noise, describing the distribution of the system conditioned on (the past of) a common noise realisation.