Probabilistic Representations of Solutions of the Forward Equations

In this paper we prove a stochastic representation for solutions of the evolution equation $ \partial_t \psi_t = {1/2}L^*\psi_t $ where $ L^* $ is the formal adjoint of an elliptic second order differential operator with smooth coefficients corresponding to the infinitesimal generator of a finite dimensional diffusion $ (X_t).$ Given $ \psi_0 = \psi $, a distribution with compact support, this representation has the form $ \psi_t = E(Y_t(\psi))$ where the process $ (Y_t(\psi))$ is the solution of a stochastic partial differential equation connected with the stochastic differential equation for $ (X_t) $ via Ito's formula.