Probabilistic representations of solutions of the forward equations

In this paper we prove a stochastic representation for solutions of the evolution equation ∂t ψt= ½L* ψt where L* is the formal adjoint of a second order elliptic differential operator L, with smooth coefficients, corresponding to the infinitesimal generator of a finite dimensional diffusion (Xt). Given ψ0= ψ, a distribution with compact support, this representation has the form ψt = E(Yt(ψ)) where the process (Yt(ψ)) is the solution of a stochastic partial differential equation connected with the stochastic differential equation for (Xt) via Ito's formula