One-dimensional diffusion processes with moving membrane: partial reflection in combination with jump-like exit of process from membrane

TL;DR

This paper constructs a Markov process with a moving membrane on a line, combining diffusion with nonlocal boundary conditions to model partial reflection and jump-like exits at the membrane.

Contribution

It introduces a novel analytical framework for Markov processes with a moving membrane incorporating partial reflection and jump-like exits via Feller semigroups.

Findings

Explicit construction of the Feller semigroup for the process.

Characterization of the process behavior at the moving membrane.

Integration of nonlocal boundary conditions into diffusion models.

Abstract

We use analytical methods to construct the two-parameter Feller semigroup associated with a Markov process on a line with a moving membrane such that at the points on both sides of the membrane it coincides with the ordinary diffusion processes given there and its behavior after reaching the membrane is described by a kind of nonlocal Feller-Wentzell conjugation condition.