Uniqueness for nonlinear Fokker–Planck equations and for McKean–Vlasov SDEs: The degenerate case

Abstract

This work is concerned with the existence and uniqueness of generalized (mild or distributional) solutions to (possibly degenerate) Fokker–Planck equations ρt−Δβ(ρ)+div(Db(ρ)ρ)=0 in (0,∞)×Rd, ρ(0,x)≡ρ0(x). Under suitable assumptions on β:R→R,b:R→R and D:Rd→Rd, d≥1, this equation generates a unique flow ρ(t)=S(t)ρ0:[0,∞)→L1(Rd) as a mild solution in the sense of nonlinear semigroup theory. This flow is also unique in the class of L∞((0,T)×Rd)∩L∞((0,T);H−1), ∀T>0, Schwartz distributional solutions on (0,∞)×Rd. Moreover, for ρ0∈L1(Rd)∩H−1(Rd), t→S(t)ρ0 is differentiable from the right on [0,∞) in H−1(Rd)-norm. As a main application, the weak uniqueness of the corresponding McKean–Vlasov SDEs is proven.