Probabilistic Representations of Solutions of the Forward Equations

In this paper we prove a stochastic representation for solutions of the evolution equation 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 (X _t ). Given ψ ₀ = ψ, a distribution with compact support, this representation has the form ψ _t  = E(Y _t (ψ)) where the process (Y _t (ψ)) is the solution of a stochastic partial differential equation connected with the stochastic differential equation for (X _t ) via Ito’s formula.