Friday, July 12, 2024 - 14:15 in V3-201+Zoom
Strong solutions to a class of degenerate McKean—Vlasov SDEs with coefficients of Nemytskii-type
A talk in the Bielefeld Stochastic Afternoon series by
Sebastian Grube from Bielefeld University
| Abstract: |
While the nondegenerate case is well-known, there are only few results on the existence of strong solutions to
McKean—Vlasov SDEs with coefficients of Nemytskii-type in the degenerate case.
We consider a broad class of degenerate nonlinear Fokker–Planck(–Kolmogorov) equations with coefficients of Nemytskii-type. This includes, in particular, the classical porous medium equation perturbed by a first-order term with initial datum in a subset of probability densities, which is dense with respect to the topology inherited from L1, and, in the one-dimensional setting, the classical porous medium equation with initial datum in an arbitrary point x in the real numbers. For these kind of equations the existence of a Schwartz-distributional solution u is well-known. We show that there exists a unique strong solution to the associated degenerate McKean–Vlasov SDE with time marginal law densities u. In particular, every weak solution to this equation with time marginal law densities u can be written as a functional of the driving Brownian motion. Moreover, plugging any Brownian motion into this very functional yields a weak solution with time marginal law densities u.
Within the CRC this talk is associated to the project(s): A5, B1 |
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