# Sticky Brownian Motion and its Numerical Solution

## TL;DR

This paper introduces a new, efficient simulation method for sticky Brownian motion, a diffusion process with boundary interaction, with applications in biology, materials science, and finance, and demonstrates significant speed improvements over existing methods.

## Contribution

It presents a simple sticky random walk model for simulating sticky Brownian motion, offering substantial computational speed gains and insights into its properties.

## Key findings

- Sticky random walk is 100 to 10,000 times faster than existing methods.
- The model accurately captures mesoscale particle behavior near boundaries.
- Potential extension to multi-dimensional sticky diffusions is discussed.

## Abstract

Sticky Brownian motion is the simplest example of a diffusion process that can spend finite time both in the interior of a domain and on its boundary. It arises in various applications such as in biology, materials science, and finance. This article spotlights the unusual behavior of sticky Brownian motions from the perspective of applied mathematics, and provides tools to efficiently simulate them. We show that a sticky Brownian motion arises naturally for a particle diffusing on $\mathbb{R}_+$ with a strong, short-ranged potential energy near the origin. This is a limit that accurately models mesoscale particles, those with diameters $\approx 100$nm-$10\mu$m, which form the building blocks for many common materials. We introduce a simple and intuitive sticky random walk to simulate sticky Brownian motion, that also gives insight into its unusual properties. In parameter regimes of practical interest, we show this sticky random walk is two to five orders of magnitude faster than alternative methods to simulate a sticky Brownian motion. We outline possible steps to extend this method towards simulating multi-dimensional sticky diffusions.