  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.
