Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Stochastic Processes

  • Alan J. McKane
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_526-3

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...


Markov Process Master Equation Stochastic Differential Equation Langevin Equation Stochastic Variable 
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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Theory Group, School of Physics and AstronomyUniversity of ManchesterManchesterUK