Fokker-planck-kolmogorov Equations

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On non-uniqueness and uniqueness of solutions in finite-horizon Mean Field Games (2017, neutral): “The estimate (4.9), obtained by probabilistic methods in the Appendix, could be replaced by which can be deduced from [8] where, however, the right hand side may depend on the second derivatives of v i .”
From nonlinear Fokker–Planck equations to solutions of distribution dependent SDE (2018, background): “ i,j≤ d, (1.3) Lµtϕ(t,x) := 1 2 Xd i,j=1 aij(t,x,µt) ∂2 ∂xi∂xj ϕ(x)+ Xd i=1 bi(t,x,µt) ∂ ∂xi ϕ(x), is the corresponding Kolmogorov operator. For equations of type (1.2), we refer the reader, e.g., to [10]. We note that (1.2) is also shortly written as (1.4) ∂tµt = L ∗ µtµt with µ 0 given. Hence, if ”
McKean–Vlasov SDEs under measure dependent Lyapunov conditions (2018, neutral): “The law of such an SDE satisfies a nonlinear Fokker–Planck–Kolmogorov equation (see also [4] and more generally [3]): writing μt := L (xt) and a := 12σσ ∗ we have, for t ∈ I, 〈μt, φ〉 = 〈μ0, φ〉+ ∫ t 0 〈 μs, b(s, ·, μs)∂xφ+ tr ( a(s, ·, μs)∂(2) xφ )〉 ds ∀φ ∈ C(2) 0 (D) .”