# Diffusion in the presence of cells with semi-permeable membranes

## TL;DR

This paper analyzes the limiting behavior of diffusion processes near semi-permeable membranes, revealing metastable distributions and boundary effects in complex domain configurations.

## Contribution

It introduces a mathematical framework for understanding diffusion with asymmetric boundary interactions and small trapping parameters, extending prior models to multiple and nested domains.

## Key findings

- Metastable distributions depend on boundary parameter magnitudes.
- Asymmetric boundary behavior influences trapping and escape dynamics.
- Limiting processes are characterized for various domain configurations.

## Abstract

We consider processes that coincide with a given diffusion process except on the boundaries of a finite collection of domains. The behavior on each of the boundaries is asymmetric: the process is much more likely to enter the interior of the domain than to enter the interior of its complement, with a small parameter controlling the trapping mechanism. We describe the limiting behavior of the processes. In particular, if the parameters controlling the boundary behavior have different orders of magnitude for different domains or if the domains are nested, metastable distributions between the trapping regions are described.