Probabilistic Representations of Solutions of the Forward Equations

Abstract

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.Comment: 29 page