Ergodicity for Singular SDEs Driven by Fractional Brownian motion

Speaker's Name: Avi Mayorcas
Speaker's Affiliation: University of Bath
Speaker's Research Theme(s): Statistics and Probability, Mathematical Physics
Abstract:
We are motivated by the question, 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. Instead, in this talk we instead focus on the scale of singular coefficients, in particular the drift term. Building on recent developments in the theory of regularisation by noise (Catellier—Gubinelli 2015, 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 (which will be explained in the talk) but removes us from the Markov setting. However, by casting our problem as 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 to oscillate sufficiently fast. Joint work ongoing with Ł. Mądry.

Venue: A17