Hostname: page-component-7bb8b95d7b-qxsvm Total loading time: 0 Render date: 2024-10-04T20:22:29.693Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic models with electrical analogues

Published online by Cambridge University Press:  24 October 2008

F. P. Kelly
F. P. Kelly
Affiliation:
Emmanuel College, Cambridge
Get access Rights & Permissions [Opens in a new window]

Extract

1. Introduction. It has long been known that a connexion exists between reversible random walks and electrical networks ((2), (4) pp. 303–310, (6)). In section 2 we outline this connexion. More recently Kingman (5) has exhibited an electrical analogy for a flow model. In section 3 we show that this analogy follows from the results on random walks and we extend the analogy to cover the transient behaviour of Kingman's model. In so far as this model may be regarded as a naive description of the mechanism by which electrons move in a conductor it provides a physical explanation of the connexion between random walks and electrical networks. In section 4 we describe an invasion model which was inspired by the work of Clifford and Sudbury (1) on models to represent competition between species. Adapting their method we display the relationship of this model to random walks and hence obtain an electrical analogy for another class of stochastic processes.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 80 , Issue 1 , July 1976 , pp. 145 - 151
Copyright
Copyright © Cambridge Philosophical Society 1976

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Clifford, P. and Sudbury, A.A model for spatial conflict. Biometrika 60 (1973), 581–588.CrossRefGoogle Scholar
(2)Hunt, G. A.Some theorems concerning Brownian motion. Trans. Amer. Math. Soc. 81 (1956),294–319.CrossRefGoogle Scholar
(3)Karlin, S.Equilibrium behaviour of population genetic models with non-random mating. II. J. Appl. Probability 5 (1963), 487–566.CrossRefGoogle Scholar
(4)Kemeny, J. G., Snell, J. L. and Knapp, A. W.Denumerable Markov chains (Princeton, Van Nostrand, 1966).Google Scholar
(5)Kingman, J. F. C.Markov population processes. J. Appl. Probability 6 (1969), 1–18.CrossRefGoogle Scholar
(6)Nash-Williams, C. ST J. A.Random walk and electric currents in networks. Proc. Cambridge Philos. Soc. 55 (1959), 181–194.CrossRefGoogle Scholar