Analytical approximation to the multidimensional Fokker--Planck equation with steady state

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Title: Analytical approximation to the multidimensional Fokker--Planck equation with steady state
Authors: Martin, R
Craster, RV
Pannier, A
Kearney, MJ
Item Type: Journal Article
Abstract: The Fokker--Planck equation is a key ingredient of many models in physics, and related subjects, and arises in a diverse array of settings. Analytical solutions are limited to special cases, and resorting to numerical simulation is often the only route available; in high dimensions, or for parametric studies, this can become unwieldy. Using asymptotic techniques, that draw upon the known Ornstein--Uhlenbeck (OU) case, we consider a mean-reverting system and obtain its representation as a product of terms, representing short-term, long-term, and medium-term behaviour. A further reduction yields a simple explicit formula, both intuitive in terms of its physical origin and fast to evaluate. We illustrate a breadth of cases, some of which are `far' from the OU model, such as double-well potentials, and even then, perhaps surprisingly, the approximation still gives very good results when compared with numerical simulations. Both one- and two-dimensional examples are considered.
Issue Date: 22-Feb-2019
Date of Acceptance: 19-Dec-2018
ISSN: 1751-8113
Publisher: IOP Publishing
Journal / Book Title: Journal of Physics A: Mathematical and Theoretical
Volume: 52
Issue: 2
Copyright Statement: © 2019 IOP Publishing Ltd. riginal content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Sponsor/Funder: Engineering & Physical Science Research Council (EPSRC)
Funder's Grant Number: EP/J009636/1
Keywords: Science & Technology
Physical Sciences
Physics, Multidisciplinary
Physics, Mathematical
stochastic processes
mean reversion
Fokker-Planck equation
01 Mathematical Sciences
02 Physical Sciences
Mathematical Physics
Publication Status: Published
Article Number: 085002
Online Publication Date: 2019-01-15
Appears in Collections:Mathematics
Applied Mathematics and Mathematical Physics
Faculty of Natural Sciences

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