  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.
