Equilibria of nonlinear distorted Brownian motions

Joint work with: Viorel Barbu (Romanian Academy, Iasi)

This talk will review the connection of nonlinear Fokker-Planck-Kolmogorov (FPK) equations and McKean-Vlasov SDEs, with special emphasis on the case where the coefficients depend Nemytskii-type on the time marginal laws. A class of examples are nonlinear distorted Brownian motions. Recent results on their asymptotic behaviour, obtained through their corresponding nonlinear FPK equations, will be presented both in the non-degenerate and degenerate case.

References
Barbu, Viorel; Röckner, Michael Probabilistic representation for solutions to nonlinear Fokker-Planck equations. SIAM J. Math. Anal. 50 (2018), no. 4, 4246–4260.
Barbu, Viorel; Röckner, Michael From nonlinear Fokker-Planck equations to solutions of distribution dependent SDE. Ann. Probab. 48 (2020), no. 4, 1902–1920.
Barbu, Viorel; Röckner, Michael Solutions for nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations. J. Funct. Anal. 280 (2021), no. 7, 108926, 35 pp.
Barbu, Viorel; Röckner, Michael The invariance principle for nonlinear Fokker-Planck equations. J. Differential Equations 315 (2022), 200–221.
Barbu, Viorel, Röckner, Michael Uniqueness for nonlinear Fokker-Planck equations and for McKean-Vlasov SDEs: The degenerate case arXiv:2203.00122
Barbu, Viorel, Röckner, Michael The evolution to equilibrium of solutions to nonlinear Fokker-Planck equations arXiv:1904.08291 , to appear in Indiana Univ. Math. J., 36 pp.