Probabilistic Representations of Solutions of the Forward Equations

Abstract

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