A Stochastic Diffusion Process for the Dirichlet Distribution

Abstract EN

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.