Uniform pathwise stability of additive singular SDEs driven by fractional Brownian motion

TL;DR

This paper investigates the long-term behavior of certain stochastic differential equations driven by fractional Brownian motion, establishing well-posedness, existence of invariant measures, and conditions for exponential stability.

Contribution

It introduces a novel approach combining dissipativity and noise regularisation to analyze stability and invariant measures for singular SDEs driven by fractional Brownian motion.

Findings

Proved well-posedness of the SDEs under specified conditions.

Established existence of invariant measures for the system.

Demonstrated exponential contraction when the singular term is sufficiently small.

Abstract

We study the long-time behaviour of solutions to a class of $d$-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H \in (0,1)$. The drift consists of a dissipative Lipschitz term and a singular term of regularity $\gamma >1-1/(2H)$ in Besov-H\"older scales. We establish well-posedness and, through a Markovian enhancement, existence of an invariant measure. If the singular contribution is sufficiently small, we prove exponential contraction of solutions, and thereby, uniqueness of the invariant measure. Our methods rely on uniform pathwise estimates which utilise together the dissipativity of the drift and the regularisation effect of the noise.