Ergodic Theory for Fractional SDE with Singular Coefficients (2511.20556v1)

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) (Hairer05 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 Hairer05 and Hairer--Ohashi 07 but manage to retain the Doob--K'hashminksii provided by Hairer--Ohashi07. 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.