A stochastic process is a process which evolves randomly in time and space. The dispersion of contaminants in gases and liquids, Brownian Motion and hydrodynamic Turbulence are well known examples, though all dynamical systems are stochastic to a lesser or greater degree. The randomness can arise in a variety of ways: through an uncertainty in the initial state of the system; the equation motion of the system contains either random coefficients or forcing functions; the system amplifies small disturbances to an extent that knowledge of the initial state of the system at the micromolecular level is required for a deterministic solution (this is a feature of NonLinear Systems of which the most obvious example is hydrodynamic turbulence).

As such, a system will evolve either temporally or spatially or both in a variety of ways, and to each outcome there is assumed to exist a unique probability of occurrence. More precisely if x(t) is a random variable representing all possible outcomes of the system at some fixed time t, then x(t) is regarded as a measurable function on a given probability space and when t varies one obtains a family of random variables (indexed by t), i.e., by definition a stochastic process, or a random function x(.) or briefly x.

Just as differential equations can be used to study the behavior of deterministic processes, so they can be used to study the behavior of stochastic processes. However, the theory of *"stochastic" differential equations* is concerned with probabilistic aspects of the process the equations describe: the explicit form of the solution of the equations is often useful but not essential. More precisely, one is interested in the determination of the distribution of x(t) (the *probability density function* (pdf) of x(t) or joint distributions at several instants or alternatively one seeks averages or moments associated with the pdf. Such averages are often referred to as *ensemble averages* to distinguish them from time averages associated with some function of x(t) as the system evolves over some sufficiently large period of time. (See Ergodic Process.)

As an example of a stochastic differential equation consider the equation frequently used to represent a simple diffusion process x(t), namely,

where μ and D are nonrandom functions and W(t) is a white noise (nondifferentiable) function with the property that dW(t) is distributed normally with zero mean and variance
(dW)^{2}
= dt. Note that because W(t) is nondifferentiable, the equation of motion cannot be represented as a standard differential equation. However, the equation for P(z,t), the pdf for the occurrence of a particular value z of x(t) at time t is a parabolic partial differential equation, namely,

The equation is commonly referred to as the *Fokker-Planck equation* in physical applications. The particular class of stochastic equations in Equation (1) which includes the well-known *Langevin equation for Brownian motion* has been used extensively to model atmospheric dispersion [MacInnes and Bracco (1992)], particle dispersion in turbulent flows (see Particle Transport in Turbulent Fluids) and as an analogue equation to generate fluid velocities in turbulent flows [Pope (1990)].

#### REFERENCES

MacInnes, J. M. and Bracco, F. V. (1992) Stochastic particle dispersion modeling and the tracer-particle limit, *Phys. Fluids* A, 4, 2809-2824.

Pope, S. B. (1990) The velocity-dissipation probability density function model for turbulent flows, *Phys. Fluids A.* 2, 1437-1449.

### Useful Introductory Texts on Stochastic Processes

van Kampen, N. G. (1981) *Stochastic Processes in Physics and Chemistry*, North-Holland Publishing Company, Amsterdam.

Wax, N. (1954) *Noise and Stochastic Processes*, Dover Publications, Inc., New York.

#### References

- MacInnes, J. M. and Bracco, F. V. (1992) Stochastic particle dispersion modeling and the tracer-particle limit,
*Phys. Fluids*A, 4, 2809-2824. - Pope, S. B. (1990) The velocity-dissipation probability density function model for turbulent flows,
*Phys. Fluids A.*2, 1437-1449. DOI: 10.1063/1.857592