### Diffusion Processes Satisfying a Conservation Law Constraint

We investigate coupled stochastic differential equations governing N non-negative continuous random variables that satisfy a conservation principle. In various fields a conservation law requires that a set of fluctuating variables be non-negative and (if appropriately normalized) sum to one. As a result, any stochastic differential equation model to be realizable must not produce events outside of the allowed sample space. We develop a set of constraints on the drift and diffusion terms of such stochastic models to ensure that both the non-negativity and the unit-sum conservation law constraint are satisfied as the variables evolve in time. We investigate the consequences of the developed constraints on the Fokker-Planck equation, the associated system of stochastic differential equations, and the evolution equations of the first four moments of the probability density function. We show that random variables, satisfying a conservation law constraint, represented by stochastic diffusion processes, must have diffusion terms that are coupled and nonlinear. The set of constraints developed enables the development of statistical representations of fluctuating variables satisfying a conservation law. We exemplify the results with the bivariate beta process and the multivariate Wright-Fisher, Dirichlet, and Lochner’s generalized Dirichlet processes.

- Publication Date:

- Report Number(s):
- LA-UR-13-28548

Journal ID: ISSN 2090-3332

- Grant/Contract Number:
- AC52-06NA25396

- Type:
- Accepted Manuscript

- Journal Name:
- International Journal of Stochastic Analysis

- Additional Journal Information:
- Journal Volume: 2014; Journal ID: ISSN 2090-3332

- Publisher:
- Hindawi

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

- Sponsoring Org:
- USDOE

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Conservation; Diffusion process; Fokker-Planck equation; Statistical moment equations

- OSTI Identifier:
- 1233162

```
Bakosi, J., and Ristorcelli, J. R..
```*Diffusion Processes Satisfying a Conservation Law Constraint*. United States: N. p.,
Web. doi:10.1155/2014/603692.

```
Bakosi, J., & Ristorcelli, J. R..
```*Diffusion Processes Satisfying a Conservation Law Constraint*. United States. doi:10.1155/2014/603692.

```
Bakosi, J., and Ristorcelli, J. R.. 2014.
"Diffusion Processes Satisfying a Conservation Law Constraint". United States.
doi:10.1155/2014/603692. https://www.osti.gov/servlets/purl/1233162.
```

```
@article{osti_1233162,
```

title = {Diffusion Processes Satisfying a Conservation Law Constraint},

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

abstractNote = {We investigate coupled stochastic differential equations governing N non-negative continuous random variables that satisfy a conservation principle. In various fields a conservation law requires that a set of fluctuating variables be non-negative and (if appropriately normalized) sum to one. As a result, any stochastic differential equation model to be realizable must not produce events outside of the allowed sample space. We develop a set of constraints on the drift and diffusion terms of such stochastic models to ensure that both the non-negativity and the unit-sum conservation law constraint are satisfied as the variables evolve in time. We investigate the consequences of the developed constraints on the Fokker-Planck equation, the associated system of stochastic differential equations, and the evolution equations of the first four moments of the probability density function. We show that random variables, satisfying a conservation law constraint, represented by stochastic diffusion processes, must have diffusion terms that are coupled and nonlinear. The set of constraints developed enables the development of statistical representations of fluctuating variables satisfying a conservation law. We exemplify the results with the bivariate beta process and the multivariate Wright-Fisher, Dirichlet, and Lochner’s generalized Dirichlet processes.},

doi = {10.1155/2014/603692},

journal = {International Journal of Stochastic Analysis},

number = ,

volume = 2014,

place = {United States},

year = {2014},

month = {3}

}