CS39: Recent Advances in Stochastic Differential Equations

Organizer: Khoa Le (University of Leeds)

Regularity of the density of singular SDEs driven by fractional noise and application to McKean-Vlasov equations

Alexandre RICHARD

First, we will consider the SDE \(dX_t = b(t,X_t) dt + dB_t\), where \(b\) is a singular drift (e.g. a distribution) and \(B\) is a fractional Brownian motion. We will review some recent results on existence and uniqueness for this equation, providing criteria linking the regularity of \(b\) and the Hurst parameter \(H\) of the fractional Brownian motion. Next, we will study the time-space regularity of the conditional density of the solution in Lebesgue-Besov spaces, and also provide Gaussian bounds. Then by exploiting this regularity, we will demonstrate the existence of solutions for McKean-Vlasov equations of the form \(dY_t = \mu_t \ast b(t,Y_t) + dB_t\), where \(\mu_t\) is the law of the solution \(Y_t\), for a drift \(b\) that can be more singular than in the linear case, and chosen in the full sub-critical regime of such SDEs. Finally, we discuss uniqueness for this singular McKean-Vlasov equation. Joint work with L. Anzeletti, L. Galeati and E. Tanré.

Supercritical SDEs driven by fractional Brownian motion with divergence free drifts

Zimo Hao

We study stochastic differential equations (SDEs) driven by fractional Brownian motion, where the drift coefficient is divergence-free and supercritical with respect to scaling. Under the assumption that the drift belongs to \(L^1_t L^1_{loc}\) and has linear growth, we establish the existence of weak solutions for Lebesgue almost everywhere initial data. Furthermore, when the Hurst parameter \(H\in (0,1/2]\) and the drift lies in \(L^{1/(1-H)}_t L^{1/(1-H)}_{loc}\), we give weak uniqueness. We also obtain the stability of the solution’s law with respect to the drift. These results, in particular, allow us to treat McKean–Vlasov SDEs. This work is part of an ongoing collaboration with Lucio Galeati.