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.
Highlights
We develop a Fokker-Planck equation whose statistically stationary solution is the Dirichlet distribution [1, 2, 3]
We show that the statistically stationary solution of Eq (2) is the Dirichlet distribution, Eq (1), provided the stochastic differential equations (SDE) coefficients satisfy b1 κ1
Using a MonteCarlo simulation we show that the statistically stationary solution of the Fokker-Planck equation (6) with drift and diffusion (10–11) is a Dirichlet distribution, Eq (1)
Summary
We develop a Fokker-Planck equation whose statistically stationary solution is the Dirichlet distribution [1, 2, 3]. A Monte Carlo solution is used to verify that the invariant distribution is Dirichlet. The Dirichlet distribution is a statistical representation of non-negative 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]
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