Ergodic Theory for Fractional SDE with Singular Coefficients

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 $B^\alpha_{\infty,\infty}(\mathbb{R}^d)$ distribution in the class treated by Catellier--Gubinelli `16, i.e. $\alpha>1-\frac{1}{2H}$. 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 L\^e `20, which allows us to efficiently treat small and large scales simultaneously. By tuning the local scale as a function of $\|g\|_{B^\alpha_{\infty,\infty}}$ we are able to obtain the necessary continuity of the semi-group and stability estimates to show unique ergodicity for all $g\in B^{\alpha}_{\infty,\infty}(\mathbb{R}^d)$. We believe that these local-global sewing lemmas may be of independent interest.