A stochastic diffusion process for Lochner's generalized Dirichlet distribution

Abstract

The method of potential solutions of Fokker-Planck equations is used to develop a transport equation for the joint probability of N stochastic variables with Lochner's generalized Dirichlet distribution (R.H. Lochner, A Generalized Dirichlet Distribution in Bayesian Life Testing, Journal of the Royal Statistical Society, Series B, 37(1):pp. 103-113, 1975) as its asymptotic solution. Individual samples of a discrete ensemble, obtained from the system of stochastic differential equations, equivalent to the Fokker-Planck equation developed here, satisfy a unit-sum constraint at all times and ensure a bounded sample space, similarly to the process developed in (J. Bakosi, J.R. Ristorcelli, A stochastic diffusion process for the Dirichlet distribution, Int. J. Stoch. Anal., Article ID, 842981, 2013) for the Dirichlet distribution. Consequently, the generalized Dirichlet diffusion process may be used to represent realizations of a fluctuating ensemble of N variables subject to a conservation principle. Compared to the Dirichlet distribution and process, the additional parameters of the generalized Dirichlet distribution allow a more general class of physical processes to be modeled with a more general covariance matrix.