## A Stochastic Diffusion Process for the Dirichlet Distribution

## Abstract

The method of potential solutions of Fokker-Planck equations is used to develop a transport equation for the joint probability ofNcoupled 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 ofNvariables 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.

- Authors:

- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1198484

- Alternate Identifier(s):
- OSTI ID: 1233159

- Report Number(s):
- LA-UR-12-26980

Journal ID: ISSN 2090-3332

- Grant/Contract Number:
- AC52-06NA25396

- Resource Type:
- Published Article

- Journal Name:
- International Journal of Stochastic Analysis

- Additional Journal Information:
- Journal Volume: 2013; Journal ID: ISSN 2090-3332

- Publisher:
- Hindawi

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Fokker-Planck equation; Stochastic diffusion; Dirichlet distribution; Monte Carlo simulation

### Citation Formats

```
Bakosi, J., and Ristorcelli, J. R. A Stochastic Diffusion Process for the Dirichlet Distribution. United States: N. p., 2013.
Web. doi:10.1155/2013/842981.
```

```
Bakosi, J., & Ristorcelli, J. R. A Stochastic Diffusion Process for the Dirichlet Distribution. United States. doi:10.1155/2013/842981.
```

```
Bakosi, J., and Ristorcelli, J. R. Fri .
"A Stochastic Diffusion Process for the Dirichlet Distribution". United States. doi:10.1155/2013/842981.
```

```
@article{osti_1198484,
```

title = {A Stochastic Diffusion Process for the Dirichlet Distribution},

author = {Bakosi, J. and Ristorcelli, J. R.},

abstractNote = {The method of potential solutions of Fokker-Planck equations is used to develop a transport equation for the joint probability ofNcoupled 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 ofNvariables 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.},

doi = {10.1155/2013/842981},

journal = {International Journal of Stochastic Analysis},

number = ,

volume = 2013,

place = {United States},

year = {2013},

month = {3}

}

DOI: 10.1155/2013/842981