Sticky Brownian Motion and its Numerical Solution

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.

References in corpus (4)

• Numerical computation of rare events via large deviation theory
• Extended law of corresponding states for protein solutions
• Multicomponent adhesive hard sphere models and short-ranged attractive interactions in colloidal or micellar solutions
• Sticky-sphere clusters

Cited by in corpus (14)

• Optimization in First-Passage Resetting
• Dynamical large deviations of reflected diffusions
• Optimization and Growth in First-Passage Resetting
• Markov Chain Approximation of One-Dimensional Sticky Diffusions
• Brownian dynamics simulations of hard rods in external fields and with contact interactions
• Close encounters of the sticky kind: Brownian motion at absorbing boundaries
• Trapping of an active Brownian particle at a partially absorbing wall
• Simulating sticky particles: A Monte Carlo method to sample a stratification
• Fast Brownian cluster dynamics
• Brownian dynamics of Dirac fermions in twisted bilayer graphene
• Functional convergence to the local time of a sticky diffusion
• The Sticky Lévy Process as a solution to a Time Change Equation
• Functional Inequalities for doubly weighted Brownian Motion with Sticky-Reflecting Boundary Diffusion
• Pricing and hedging for a sticky diffusion