A Stochastic Diffusion Process for the Dirichlet Distribution
Abstract
The method of potential solutions of FokkerPlanck equations is used to develop a transport equation for the joint probability of N coupled 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 unitsum constraint at all times. The process may be used to represent realizations of a fluctuating ensemble of N variables subject to a conservation principle. Similar to the multivariate WrightFisher 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:

 Los Alamos National Laboratory, Los Alamos, NM 87545, USA
 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):
 LAUR1226980
Journal ID: ISSN 20903332; PII: 842981; 842981
 Grant/Contract Number:
 AC5206NA25396
 Resource Type:
 Published Article
 Journal Name:
 International Journal of Stochastic Analysis
 Additional Journal Information:
 Journal Name: International Journal of Stochastic Analysis Journal Volume: 2013; Journal ID: ISSN 20903332
 Publisher:
 Hindawi Publishing Corporation
 Country of Publication:
 Egypt
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; FokkerPlanck equation; Stochastic diffusion; Dirichlet distribution; Monte Carlo simulation
Citation Formats
Bakosi, J., and Ristorcelli, J. R. A Stochastic Diffusion Process for the Dirichlet Distribution. Egypt: N. p., 2013.
Web. doi:10.1155/2013/842981.
Bakosi, J., & Ristorcelli, J. R. A Stochastic Diffusion Process for the Dirichlet Distribution. Egypt. doi:10.1155/2013/842981.
Bakosi, J., and Ristorcelli, J. R. Wed .
"A Stochastic Diffusion Process for the Dirichlet Distribution". Egypt. 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 FokkerPlanck equations is used to develop a transport equation for the joint probability of N coupled 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 unitsum constraint at all times. The process may be used to represent realizations of a fluctuating ensemble of N variables subject to a conservation principle. Similar to the multivariate WrightFisher 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 = {Egypt},
year = {2013},
month = {4}
}
DOI: 10.1155/2013/842981
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