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Title: A Stochastic Diffusion Process for the Dirichlet Distribution

Abstract

The method of potential solutions of Fokker-Planck equations is used to develop a transport equation for the joint probability ofNcoupled stochastic variables with the Dirichlet distribution as its asymptotic solution. To ensure a bounded sample space, a coupled nonlinear diffusion process is required: the Wiener processes in the equivalent system of stochastic differential equations are multiplicative with coefficients dependent on all the stochastic variables. Individual samples of a discrete ensemble, obtained from the stochastic process, satisfy a unit-sum constraint at all times. The process may be used to represent realizations of a fluctuating ensemble ofNvariables subject to a conservation principle. Similar to the multivariate Wright-Fisher process, whose invariant is also Dirichlet, the univariate case yields a process whose invariant is the beta distribution. As a test of the results, Monte Carlo simulations are used to evolve numerical ensembles toward the invariant Dirichlet distribution.

Authors:
 [1];  [1]
  1. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Publication Date:
Research Org.:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1198484
Alternate Identifier(s):
OSTI ID: 1233159
Report Number(s):
LA-UR-12-26980
Journal ID: ISSN 2090-3332
Grant/Contract Number:  
AC52-06NA25396
Resource Type:
Published Article
Journal Name:
International Journal of Stochastic Analysis
Additional Journal Information:
Journal Volume: 2013; Journal ID: ISSN 2090-3332
Publisher:
Hindawi
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Fokker-Planck equation; Stochastic diffusion; Dirichlet distribution; Monte Carlo simulation

Citation Formats

Bakosi, J., and Ristorcelli, J. R. A Stochastic Diffusion Process for the Dirichlet Distribution. United States: N. p., 2013. Web. doi:10.1155/2013/842981.
Bakosi, J., & Ristorcelli, J. R. A Stochastic Diffusion Process for the Dirichlet Distribution. United States. doi:10.1155/2013/842981.
Bakosi, J., and Ristorcelli, J. R. Fri . "A Stochastic Diffusion Process for the Dirichlet Distribution". United States. doi:10.1155/2013/842981.
@article{osti_1198484,
title = {A Stochastic Diffusion Process for the Dirichlet Distribution},
author = {Bakosi, J. and Ristorcelli, J. R.},
abstractNote = {The method of potential solutions of Fokker-Planck equations is used to develop a transport equation for the joint probability ofNcoupled stochastic variables with the Dirichlet distribution as its asymptotic solution. To ensure a bounded sample space, a coupled nonlinear diffusion process is required: the Wiener processes in the equivalent system of stochastic differential equations are multiplicative with coefficients dependent on all the stochastic variables. Individual samples of a discrete ensemble, obtained from the stochastic process, satisfy a unit-sum constraint at all times. The process may be used to represent realizations of a fluctuating ensemble ofNvariables subject to a conservation principle. Similar to the multivariate Wright-Fisher process, whose invariant is also Dirichlet, the univariate case yields a process whose invariant is the beta distribution. As a test of the results, Monte Carlo simulations are used to evolve numerical ensembles toward the invariant Dirichlet distribution.},
doi = {10.1155/2013/842981},
journal = {International Journal of Stochastic Analysis},
number = ,
volume = 2013,
place = {United States},
year = {2013},
month = {3}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
DOI: 10.1155/2013/842981

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