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

Let [Formula: see text] be the space of probability measures on [Formula: see text] with finite second moment. The path independence of additive functionals of McKean–Vlasov SDEs is characterized by PDEs on the product space [Formula: see text] 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 [Formula: see text] 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.

Girsanov Transformations for Non-Symmetric Diffusions

Let X be a diffusion process, which is assumed to be associated with a (non-symmetric) strongly local Dirichlet form (ℰ, 𝓓 (ℰ)) on L2(E ;m). For u ∈ 𝓓(ℰ)e, the extended Dirichlet space, we investigate some properties of the Girsanov transformed process Y of X . First, let be the dual process of X and Ŷ the Girsanov transformed process of . We give a necessary and sufficient condition for (Y , Ŷ to be in duality with respect to the measure e2um. We also construct a counterexample, which shows that this condition may not be satisfied and hence (Y , Ŷ ) may not be dual processes. Then we present a sufficient condition under which Y is associated with a semi-Dirichlet form. Moreover, we give an explicit representation of the semi-Dirichlet form.