Probabilistic representations of solutions of the forward equations

Abstract

In this paper we prove a stochastic representation for solutions of the evolution equation ∂_t ψ_t= ½L* ψ_t 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 ψ₀= 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.