Probabilistic Representations of Solutions of the Forward Equations

Abstract

In this paper we prove a stochastic representation for solutions of the evolution equation ${\partial }_t{\psi }_t=1/2L^{\ast }{\psi }_t$ where $L^{\ast }$ 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.

29 pages