Space-distribution PDEs for path independent additive functionals of McKean-Vlasov SDEs

Let P₂ (ℝ^d) be the space of probability measures on ℝ^d with finite second moment. The path independence of additive functionals of McKean-Vlasov SDEs is characterized by PDEs on the product space ℝ^d × P₂ (ℝ^d) equipped with the usual derivative in space variable and Lions' derivative in distribution. These PDEs are solved by using probabilistic arguments developed from Ref. 2. As a consequence, the path independence of Girsanov transformations is identified with nonlinear PDEs on ℝ^d × P₂ (ℝ^d) whose solutions are given by probabilistic arguments as well. In particular, the corresponding results on the Girsanov transformation killing the drift term derived earlier for the classical SDEs are recovered as special situations.