Uniqueness for nonlinear Fokker–Planck equations and for McKean–Vlasov SDEs: The degenerate case
نویسندگان
چکیده
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 D:Rd→Rd, d≥1, this equation generates a unique flow ρ(t)=S(t)ρ0:[0,∞)→L1(Rd) as mild solution sense nonlinear semigroup theory. also class L∞((0,T)×Rd)∩L∞((0,T);H−1), ∀T>0, Schwartz distributional (0,∞)×Rd. Moreover, for ρ0∈L1(Rd)∩H−1(Rd), t→S(t)ρ0 differentiable from right [0,∞) H−1(Rd)-norm. As main application, weak corresponding McKean–Vlasov SDEs proven.
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ژورنال
عنوان ژورنال: Journal of Functional Analysis
سال: 2023
ISSN: ['0022-1236', '1096-0783']
DOI: https://doi.org/10.1016/j.jfa.2023.109980