El Mehdi Haress (University of Leeds) – Singular and dissipative SDEs

 In this talk, I will present recent results on the long-time stability of additive stochastic differential equations driven by fractional Brownian motion. The drift is decomposed into a singularity (that can be a geniune distribution) and a Lipschitz dissipativity. I will begin with motivations and examples where such equations naturally arise, before introducing the main analytical ideas used in the proofs, in particular regularisation by noise and comparison with the Ornstein–Uhlenbeck process. These tools allow us to establish a uniform-in-time bound on the moments of the solution together with a stability result with respect to the initial condition.
I will then briefly recall a numerical scheme for approximating solutions, which will serve as a basis to discuss ongoing projects and open questions, including coupling arguments under general dissipativity assumptions, approximation of Gibbs measures, and numerical approximations for the stochastic Allen–Cahn equation with singular drift.