Sticky Brownian Motion and Its Numerical Solution

Nawaf Bou-Rabee, Miranda C. Holmes-Cerfon

Research output: Contribution to journalArticlepeer-review

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 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 R+ with a strong, short-ranged potential energy near the origin. This is a limit that accurately models mesoscale particles, those with diameters_100nm-10_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.

Original languageEnglish (US)
Pages (from-to)164-195
Number of pages32
JournalSIAM Review
Volume62
Issue number1
DOIs
StatePublished - 2020

Keywords

  • Feller boundary condition
  • Finite difference methods
  • Fokker{planck equation
  • Generalized wentzell boundary condition
  • Kolmogorov equation
  • Markov chain approximation method
  • Markov jump process
  • Sticky brownian motion
  • Sticky random walk

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Mathematics
  • Applied Mathematics

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