Probabilistic Representations of Solutions of the Forward Equations

TL;DR

This paper establishes a stochastic representation for solutions of a class of forward evolution equations associated with elliptic differential operators, linking PDE solutions to stochastic processes through Ito's formula.

Contribution

It provides a novel stochastic representation for solutions of forward equations involving elliptic operators, connecting PDE solutions with stochastic differential equations.

Findings

Representation of solutions as expected values of stochastic processes

Connection between PDE solutions and stochastic differential equations via Ito's formula

Framework applicable to diffusion processes with smooth coefficients

Abstract

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