"Well-posedness of McKean-Vlasov SDEs with density-dependent drift"
A talk in the Bielefeld Stochastic Afternoon series by
Anh-Dung Le
| Abstract: | In this paper, we study the well-posedness of McKean-Vlasov SDEs whose
drift depends pointwisely on marginal density and satisfies a
condition about local integrability in time-space variables. The drift
is also assumed to be Lipschitz continuous in distribution variable
with respect to Wasserstein metric $W_p$. Our approach is by
approximation with mollified SDEs. We recall and establish stability
estimates from which we deduce that the marginal distributions (resp.
marginal densities) of the mollified SDEs converge in $W_p$ (resp.
topology of compact convergence) to the solution of the Fokker-Planck
equation associated with the SDE of interest. The weak existence of a
solution follows from an application of superposition principle. We
also prove the strong existence of a solution. The weak and strong
uniqueness are obtained in case the drift coefficient is bounded and
the diffusion coefficient is distribution-free. Within the CRC this talk is associated to the project(s): A5, B1 |