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

TL;DR

<3-5 sentence high-level summary> This work analyzes SDEs in $\mathbb{R}^d$ driven by fractional Brownian motion with a dissipative Lipschitz drift and a potentially singular drift in Besov--Hölder spaces. By combining regularisation by noise with a Markovian enhancement that incorporates the noise history, the authors obtain well-posedness results (weak for broad $\gamma$, strong for $\gamma>1-1/(2H)$ with Lipschitz $F$) and construct invariant measures. They prove uniform-in-time moment bounds and, when the irregular drift is sufficiently small, exponential contraction which yields uniqueness of the invariant measure. The framework provides a direct approach to long-time behavior for non-Markovian dynamics and offers a foundation for ergodic analysis and long-time numerical schemes in irregular settings.

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 $γ>1-1/(2H)$ in Besov-Hölder 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.

Theorems & Definitions (40)

• Theorem
• Definition 2.1
• Theorem 3.1
• Theorem 3.2
• Proposition 3.3
• proof
• Proposition 3.4
• proof
• Proposition 3.5
• proof