A Stochastic Diffusion Process for the Dirichlet Distribution. (10th April 2013)

• Record Type: Journal Article
• Authors: Bakosi, J.; Ristorcelli, J. R.
• Other Names: Xu Hong K. Academic Editor.
• Is Part Of: International journal of stochastic analysis. Volume 2013(2013)
• Publication Date: 2013-04-10
• Subjects: Stochastic analysis -- Periodicals; Stochastic analysis; Periodicals; 519.22
• DOI: 10.1155/2013/842981
• Languages: English
• ISSNs: 2090-3332

Abstract

Abstract : 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.