SIAM Review, Volume 62, Issue 1, Page 164-195, January 2020.
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 in fields such as 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, which also gives insight into its unusual properties. In parameter regimes of practical interest, we show that 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 toward simulating multidimensional sticky diffusions.

中文翻译：

---

粘性布朗运动及其数值解

SIAM评论，第62卷，第1期，第164-195页，2020年1月。
粘性布朗运动是扩散过程的最简单示例，该过程可以在域内部及其边界上花费有限的时间。它出现在生物学，材料科学和金融等领域的各种应用中。本文从应用数学的角度重点介绍了粘性布朗运动的异常行为，并提供了有效模拟它们的工具。我们表明，对于在$ \ mathbb {R} _ + $上扩散的粒子，在原点附近具有很强的短距离势能，自然会产生粘性布朗运动。这是一个精确建模中尺度粒子的限制，中尺度粒子的直径为$ \大约100 $ nm $ -10 \ mu $ m，是许多常见材料的基础。我们介绍了一种简单直观的粘性随机游动，以模拟粘性布朗运动，这也可以洞察其不寻常的特性。在实际感兴趣的参数体制中，我们证明了这种粘性随机游走比模拟粘性布朗运动的替代方法快两到五个数量级。我们概述了可能的步骤，以将该方法扩展为模拟多维粘性扩散。