  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.
