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In this paper we prove a stochastic representation for solutions of theevolution equation $ \partial_t \psi_t = {1/2}L^*\psi_t $ where $ L^* $ is theformal adjoint of an elliptic second order differential operator with smoothcoefficients corresponding to the infinitesimal generator of a finitedimensional diffusion $ (X_t).$ Given $ \psi_0 = \psi $, a distribution withcompact support, this representation has the form $ \psi_t = E(Y_t(\psi))$where the process $ (Y_t(\psi))$ is the solution of a stochastic partialdifferential equation connected with the stochastic differential equation for $(X_t) $ via Ito's formula.

Probabilistic Representations of Solutions of the Forward Equations