TY - JOUR
T1 - Sticky Brownian Motion and Its Numerical Solution
AU - Bou-Rabee, Nawaf
AU - Holmes-Cerfon, Miranda C.
N1 - Funding Information:
∗Received by the editors June 14, 2019; accepted for publication (in revised form) August 12, 2019; published electronically February 11, 2020. https://doi.org/10.1137/19M1268446 Funding: The work of the first author was partially supported by the NSF under grant DMS-181637. The work of the second author was partially supported by Department of Energy grant DE-SC0012296 and the Alfred P. Sloan Foundation. †Department of Mathematical Sciences, Rutgers University, Camden, NJ 08102 (nawaf.bourabee@ rutgers.edu). ‡Courant Institute of Mathematical Sciences, New York University, New York, NY 10012-1185 (holmes@cims.nyu.edu).
Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics Publications. All rights reserved.
PY - 2020
Y1 - 2020
N2 - 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.
AB - 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.
KW - Feller boundary condition
KW - Finite difference methods
KW - Fokker{planck equation
KW - Generalized wentzell boundary condition
KW - Kolmogorov equation
KW - Markov chain approximation method
KW - Markov jump process
KW - Sticky brownian motion
KW - Sticky random walk
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U2 - 10.1137/19M1268446
DO - 10.1137/19M1268446
M3 - Article
AN - SCOPUS:85088899922
SN - 0036-1445
VL - 62
SP - 164
EP - 195
JO - SIAM Review
JF - SIAM Review
IS - 1
ER -