A Stochastic Diffusion Process for the Dirichlet Distribution

The method of potential solutions of Fokker-Planck 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 requ...

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Bibliographic Details
Published in:International Journal of Stochastic Analysis 2013, Vol.2013 (2013), p.166-172
Main Authors: Bakosi, J., Ristorcelli, J. R.
Format: Article
Language:English
Subjects:
Asymptotic properties
Computer simulation
Diffusion
Dirichlet distribution
Dirichlet problem
Distribution (Probability theory)
Fokker-Planck equation
Invariants
Mathematical analysis
Mathematical models
Mathematical research
MATHEMATICS AND COMPUTING
Monte Carlo simulation
Series, Dirichlet
Stochastic diffusion
Stochastic processes
Stochasticity
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Summary:The method of potential solutions of Fokker-Planck 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 unit-sum 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 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.
ISSN:2090-3332
2090-3340
DOI:10.1155/2013/842981