Probabilistic Representations of Solutions of the Forward Equations By

Abstract: In this paper we prove a stochastic representation for solutions of the evolution equation ∂tψt = 1 2 L∗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. Key words: Stochastic differential equation, Stochastic partial differen-tial equation, evolution equation, stochastic flows, Ito’s formula, stochastic representation, adjoints, diffusion processes, second order elliptic partial dif-ferential equation, monotonicity inequality.