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Stochastic Processes

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

The most common type of stochastic process comprises of a set of random variables {x(t)}, where t represents the time which may be real or integer valued. Other types of stochastic process are possible, for instance when the stochastic variable depends on the spatial position r, as well as, or instead of, t. Since in the study of complex systems we will predominantly be interested in applications relating to stochastic dynamics, we will suppose that it depends only on t. One of the earliest investigations of a stochastic process was carried out by Bachelier (1900), who used the idea of a random walk to analyze stock market fluctuations. The problem of a random walk was more generally discussed by Pearson (1905) and applied to the investigation of Brownian motion by Einstein (1905, 1906), Smoluchowski (1906) and Langevin (1908). The example of a random walk illustrates the fact that in addition to time being discrete or continuous, the stochastic variable...

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Abbreviations

Fokker-Planck equation:

A partial differential equation of the second order for the time evolution of the probability density function of a stochastic process. It resembles a diffusion equation, but has an extra term that represents the deterministic aspects of the process.

Langevin equation:

A stochastic differential equation of the simplest kind: linear and with an additive Gaussian white noise. Introduced by Langevin (1908) in 1908 to describe Brownian motion; many stochastic differential equations in physics go by this name.

Markov process:

A stochastic process in which the current state of the system is only determined from its state in the immediate past, and not by its entire history.

Markov chain:

A Markov process where both the states and the time are discrete and where the process is stationary.

Master equation:

The equation describing a continuous-time Markov chain.

Stochastic process:

A sequence of stochastic variables. This sequence is usually a time-sequence, but could also be spatial.

Stochastic variable:

A random variable. This is a function that maps outcomes to numbers (real or integer).

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Authors and Affiliations

  1. Theory Group, School of Physics and Astronomy, University of Manchester, Manchester, UK

    Alan J. McKane

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  1. Alan J. McKane
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Correspondence to Alan J. McKane .

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  1. Ramtech Ltd., Larkspur, California, USA

    Robert A. Meyers

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McKane, A.J. (2015). Stochastic Processes. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27737-5_526-3

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  • DOI: https://doi.org/10.1007/978-3-642-27737-5_526-3

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