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. 1. Objective We develop a Fokker-Planck equation whose statistically stationary solution is the Dirichlet distribution [1–3]. The system of stochastic differential equations (SDE), equivalent to the Fokker-Planck equation, yields a Markov process that allows a Monte Carlo method to numerically evolve an ensemble of fluctuating variables that satisfy a unit-sum requirement. A Monte Carlo solution is used to verify that the invariant distribution is Dirichlet. The Dirichlet distribution is a statistical representation of nonnegative variables subject to a unit-sum requirement. The properties of such variables have been of interest in a variety of fields, including evolutionary theory [4], Bayesian statistics [5], geology [6, 7], forensics [8], econometrics [9], turbulent combustion [10], and population biology [11]. 2. Preview of Results The Dirichlet distribution [1–3] for a set of scalars , , , is given by where are parameters, , and denotes the gamma function. We derive the stochastic diffusion process, governing the scalars, , where is an isotropic vector-valued Wiener process [12], and , , and are coefficients. We show that the statistically stationary solution of (2) is the Dirichlet distribution, (1), provided that the SDE coefficients satisfy The restrictions imposed on the SDE coefficients, , , and , ensure reflection towards the interior of the sample space, which is a generalized triangle or tetrahedron (more precisely, a simplex) in dimensions. The restrictions together with the specification of the Nth scalar as ensure Indeed, inspection of
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