Distribution dependent SDEs driven by additive continuous and fractional Brownian noise

Abstract: Distribution dependent SDEs (or McKean—Vlasov equations) are important from both the point of view of mathematical analysis and applications; in the case of Brownian noise they are closely related to nonlinear parabolic PDEs.

In this talk I will present some recent joint work with L. Galeati & F. Harang, in which we prove a variety of well-posedness results for McKean—Vlasov equations driven by either additive continuous or fractional Brownian noise. In the former case we extend some of the recent results by Coghi, Deuschel, Friz & Maurelli to non-Lipschitz drifts, establishing separate criteria for existence and uniqueness and providing a small extension of known propagation of chaos results. However, since our results in this case also apply for zero noise they do cannot make use of any regularisation effects; in contrast, for McKean—Vlasov equations driven by fBm we extend the results of Catellier & Gubinelli for SDEs driven by fBm to the distribution dependent setting. We are able to treat McKean—Vlasov equations with singular drifts provided the dynamics are driven by an additive fBm of suitably low Hurst parameter.

differential geometryprobability

Audience: advanced learners

Young Researchers between Geometry and Stochastic Analysis 2021