Stochastic Analysis Seminar – Avi Mayorcas (University of Bath)

When does a stochastic differential equation have a unique invariant measure? In the case of bounded and measurable coefficients with Brownian noise this question is answered by Hörmander-1967. In this talk instead, we focus on the effect of singular coefficients, in particular the drift term. Building on recent developments in the theory of regularisation by noise (Catellier—Gubinelli 2015 to Galeati—Gerenscér 2025) we consider a stochastic differential equation with distributional drift, linear confinement and additive fractional Brownian noise. The fractional Brownian motion provides a regularising effect but removes us from the Markov setting. However, by casting our problem as a stochastic dynamical system (Hairer 2005 and Hairer—Ohashi 2007) we are able to show existence and uniqueness of the invariant measure for drifts of arbitrary irregularity provided the noise is chosen of sufficiently low Hurst parameter. We compare the (ir)regularity requirements imposed on the drift by our result with known no-go results in the one dimensional case.