Flow selections for (nonlinear) Fokker-Planck-Kolmogorov equations

Abstract

We provide a method to select flows of solutions to the Cauchy problem for linear and nonlinear Fokker-Planck-Kolmogorov equations (FPK equations) for measures on Euclidean space. In the linear case, our method improves similar results of a previous work of the author. Our consideration of flow selections for nonlinear equations, including the particularly interesting case of Nemytskiitype coefficients, seems to be new. We also characterize the (restricted) well-posedness of FPK equations by the uniqueness of such (restricted) flows. Moreover, we show that under suitable assumptions in the linear case such flows are Markovian, i.e. they fulfill the Chapman-Kolmogorov equations.

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

“…In particular, by (3.22), it follows that Hence, if ρ 0 is nonnegative and ρ ≥ 0, a.e. on (0, T ) × R d , it follows by Lemma 2.3 in [24] that there exists a dt ⊗ dt-version ρ of ρ such that [0, T ] ∋ t → ρ(t, x)dx is narrowly continuous and ρ(0, x)dx = ρ 0 (dx). Remark 3.3 then implies the following consequence of Theorem 3.2.…”