On nonlinear Markov processes in the sense of McKean

The invariance principle for nonlinear Fokker–Planck equations

“…The uniqueness proof (Theorem 4.3) is based on Theorems 3.1 and 3.3. As a consequence of this and Corollary 3.8 in [18], we finally obtain Theorem 4.4, which in particular realizes McKean's vision from [16], i.e. to identify the solutions of nonlinear PDEs as the one-dimensional time marginal law densities of a nonlinear Markov process, for nonlocal PDEs as in (1.1).…”

“…(4.5), with start in s > 0 instead of zero, it follows by exactly the same arguments as in the proof of Lemma 2.12 in [23] that P = P. Proof. The assertion follows from Corollary 3.8 in [18] (see also Example (iii) in Section 4.2 of [18] for the special case with Ψ(r) := r s , s ∈ 1 2 , 1 ). Remark 4.5.…”