PUB - Publikationen an der Universität Bielefeld

This work is concerned with the existence and uniqueness of generalized (mild or distributional) solutions to (pos-sibly degenerate) Fokker-Planck equations rho t -Delta beta(rho) + div(Db(rho)rho) = 0 in (0, oo) x Rd, rho(0, x) equivalent to rho 0(x). Under suitable assumptions on beta : R-+ R, b : R-+ R and D : Rd-+ Rd, d > 1, this equation generates a unique flow rho(t) = S(t)rho 0 : [0, oo)-+ L1(Rd) as a mild solution in the sense of nonlinear semigroup theory. This flow is also unique in the class of L infinity((0, T) x Rd) n L infinity((0, T);H-1), VT > 0, Schwartz distributional solutions on (0, oo) xRd. Moreover, for rho 0 E L1(Rd) nH-1(Rd), t-+ S(t)rho 0 is differentiable from the right on [0, oo) in H-1(Rd)-norm. As a main application, the weak uniqueness of the corresponding McKean-Vlasov SDEs is proven. (c) 2023 Elsevier Inc. All rights reserved.