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Sticky Brownian motion is the simplest example of a diffusion process thatcan spend finite time both in the interior of a domain and on its boundary. Itarises in various applications such as in biology, materials science, andfinance. This article spotlights the unusual behavior of sticky Brownianmotions from the perspective of applied mathematics, and provides tools toefficiently simulate them. We show that a sticky Brownian motion arisesnaturally for a particle diffusing on $\mathbb{R}_+$ with a strong,short-ranged potential energy near the origin. This is a limit that accuratelymodels mesoscale particles, those with diameters $\approx 100$nm-$10\mu$m,which form the building blocks for many common materials. We introduce a simpleand intuitive sticky random walk to simulate sticky Brownian motion, that alsogives insight into its unusual properties. In parameter regimes of practicalinterest, we show this sticky random walk is two to five orders of magnitudefaster than alternative methods to simulate a sticky Brownian motion. Weoutline possible steps to extend this method towards simulatingmulti-dimensional sticky diffusions.

Sticky Brownian Motion and its Numerical Solution