The snapping out Brownian motion

TL;DR

This paper introduces the snapping out Brownian motion, a stochastic process that models diffusion with a semi-permeable barrier at zero, providing a probabilistic representation and simulation methods for solutions with discontinuities.

Contribution

It presents a novel probabilistic representation for diffusion equations with discontinuous solutions at zero, using the snapping out Brownian motion, and offers a simulation scheme for practical applications.

Findings

Provides a probabilistic representation of discontinuous diffusion solutions.

Studies properties of the snapping out Brownian motion.

Includes a simulation scheme for applications.

Abstract

We give a probabilistic representation of a one-dimensional diffusion equation where the solution is discontinuous at $0$ with a jump proportional to its flux. This kind of interface condition is usually seen as a semi-permeable barrier. For this, we use a process called here the snapping out Brownian motion, whose properties are studied. As this construction is motivated by applications, for example, in brain imaging or in chemistry, a simulation scheme is also provided.