Bernstein diffusions for a class of linear parabolic partial differential equations

TL;DR

This paper establishes the existence of Bernstein processes linked to linear parabolic PDEs, demonstrating their reversibility, stochastic representations, and applications to heat equations with boundary conditions.

Contribution

It introduces a novel connection between Bernstein processes and linear parabolic PDEs, including reversibility and stochastic integral representations.

Findings

Bernstein processes associated with linear parabolic PDEs exist under certain conditions.

These processes can be reversible Markov diffusions.

Feynman-Kac formulas are derived for solutions to the PDEs.

Abstract

In this article we prove the existence of Bernstein processes which we associate in a natural way with a class of linear parabolic initial-and final boundary value problems defined in bounded convex subsets of Euclidean space of arbitrary dimension. Under certain conditions regarding their endpoint distributions, we also prove that such processes become reversible Markov diffusions. Furthermore we show that those diffusions satisfy two It\^o equations for some suitably constructed Wiener processes, and from that analysis derive Feynman-Kac representations for the solutions to the given equations. We then illustrate some of our results by considering the heat equation with Neumann boundary conditions both in a one-dimensional bounded interval and in a two-dimensional disk.