Ergodic Theory for Fractional SDE with Singular Coefficients
Avi Mayorcas, Łukasz Mądry
TL;DR
"Ergodic Theory for Fractional SDE with Singular Coefficients" analyzes the long-time behavior of non-Markovian SDEs of the form $X_t = x_0 + \int_0^t (g(X_r)+u(X_r))\,dr + W^H_t$ with $W^H$ a fractional Brownian motion with $H\in(0,1/2)$ and singular drift $g$ in the Besov space $B^\alpha_{\infty,\infty}$. The authors develop a bona fide stochastic dynamical system (SDS) framework for such equations by weakening Hairer–Ohashi’s continuity assumptions and introducing flexible local–global stochastic sewing lemmas to handle multi-scale regularization and stability. They prove existence (weak) and, under sharper regularity $\alpha>1-\tfrac{1}{2H}$, strong well-posedness, leading to a unique ergodic invariant measure with Gaussian moment bounds for all $g\in B^\alpha_{\infty,\infty}$ and conclusive stability estimates. A key novelty is the Bismut–Li-type formula for the fBm-driven SDE with distributional drift, enabling Malliavin-based strong Feller properties, together with irreducibility and quasi-Markovianity to obtain unique ergodicity. The local–global sewing technique and the SDS construction may be of independent interest for irregular stochastic dynamics driven by non-Markovian Gaussian noises.
Abstract
We show existence and uniqueness of invariant measures for SDE of the form \[ dX_t = g(X_t)dt + u(X_t)dt + dW^H_t \] where $W^H$ is a fractional Brownian motion (fBm) with Hurst parameter $H\in (0,\frac{1}{2})$, $u$ is a linearly dispersive term and $g$ is any $B^α_{\infty,\infty}(\mathbb{R}^d)$ distribution in the class treated by Catellier--Gubinelli `16, i.e. $α>1-\frac{1}{2H}$. The significant challenge is to combine the regularizing effect of the fBm with an ergodic theory suited to non-Markovian SDE. Concerning the latter our first main contribution is to construct a bona fide stochastic dynamical system (SDS) (Hairer `05 and Hairer--Ohashi `07) associated to the equation above. Since the solution map is only continuous in the support of the stationary noise process we weaken the definitions introduced by Hairer `05 and Hairer--Ohashi `07 but manage to retain the Doob--K'hashminksii provided by Hairer--Ohashi `07. Our second innovation is to introduce a family of flexible local-global stochastic sewing lemmas, in the vein of Lê `20, which allows us to efficiently treat small and large scales simultaneously. By tuning the local scale as a function of $\|g\|_{B^α_{\infty,\infty}}$ we are able to obtain the necessary continuity of the semi-group and stability estimates to show unique ergodicity for all $g\in B^α_{\infty,\infty}(\mathbb{R}^d)$. We believe that these local-global sewing lemmas may be of independent interest.