Ergodic Theory for Fractional SDE with Singular Coefficients

TL;DR

This paper establishes the existence and uniqueness of invariant measures for a class of fractional SDEs with singular coefficients, combining ergodic theory, stochastic dynamical systems, and novel sewing lemmas.

Contribution

It introduces a new framework for ergodic theory of non-Markovian SDEs driven by fractional Brownian motion with singular coefficients, including a construction of stochastic dynamical systems and local-global sewing lemmas.

Findings

Proved existence and uniqueness of invariant measures for the class of fractional SDEs.

Developed a family of local-global stochastic sewing lemmas for scale analysis.

Achieved stable ergodic behavior for all singular coefficients in the specified Besov space.

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 $B^\alpha_{\infty,\infty}(\mathbb{R}^d)$ 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.