Lukasz Madry (University of Brest) – Ergodic theory for fractional SDE with singular coefficients

In this talk I will present a result on the construction of the unique invariant measure of the singular SDE with fractional Brownian noise (fBm), equipped with a linear damping. We build up on the theory of regularisation by noise, developed in recent years by Catellier, Gubinelli, Galeati and many others, and merge it with ergodic theory of fractional SDEs, studied by Hairer and his coauthors. We establish tightness in the usual regime of weak existence ($\alpha > 1/2-1/(2H)$, where $\alpha$ is Besov-Holder regularity of the drift and H is Hurst index of fBm), and uniqueness of invariant measure under usual condition on well-posedness for singular SDEs driven by fractional Brownian motion ($\alpha > 1 - 1/(2H)$). To this end we employ a modification of stochastic sewing, which also allows us to show Gaussian tails of the solution. Our approach does not require any assumption on the size of the drift with respect to the damping strength.