Sticky Brownian motions and a probabilistic solution to a two-point boundary value problem

TL;DR

This paper introduces a probabilistic approach to solving a boundary value problem for the heat equation using sticky Brownian motions, providing a novel stochastic representation for the solution.

Contribution

It establishes a new connection between boundary value problems and sticky Brownian motions, offering a probabilistic solution method for a specific class of PDEs.

Findings

Probabilistic representation of the solution via sticky Brownian motion.

Derivation of the stochastic differential equation for sticky Brownian motion.

Application to boundary conditions involving derivatives.

Abstract

In this paper, we study a two-point boundary value problem consisting of the heat equation on the open interval $(0,1)$ with boundary conditions which relate first and second spatial derivatives at the boundary points. Moreover, the unique solution to this problem can be represented probabilistically in terms of a sticky Brownian motion. This probabilistic representation is attained from the stochastic differential equation for a sticky Brownian motion on the bounded interval $[0,1]$.