Nonlinear Fokker-Planck-Kolmogorov Equations and Stochastic Distribution Dependent SDE
A talk in the Bielefeld Stochastic Afternoon series by Michael Röckner from Bielefeld
By Ito's formula the time marginals of a solution to a distribution dependent SDE solve a nonlinear Fokker-Planck-Kolmogorov equation. This talk is about the converse: we present a general technique how to identify a solution to a nonlinear Fokker-Planck-Kolmogorov equation consisting of probability densities as the time marginals of a solution to a distribution dependent SDE. We apply this to the special case of a porous media equation perturbed by the divergence of a vector field depending nonlinearly on the solution. More precisely, we construct a generalized entropic solution u to this equation and apply the above general technique to find the corresponding distribution dependent SDE which has a weak solution with marginals given by $u$. We thus gain a probabilistic representation of $u$. The final aim is to develop a general theory relating distribution dependent SDE and nonlinear Fokker-Planck-Kolmogorov equations analogous to the classical linear case.
This is joint work with Viorel Barbu (Romanian Academy, Iasi)
Reference: "Probabilistic representation of solutions to Fokker-Planck equations",
SIAM J. Math. Analysis 50 (2018), no. 4, 4246-4260 and arXiv:1801.10510
Within the CRC this talk is associated to the project(s): A5, B1, B2