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A note on random walks at constant speed

Published online by Cambridge University Press:  01 July 2016

M. S. Bartlett
M. S. Bartlett*
Affiliation:
Address: 117 Littlehampton Road, Worthing BN13 1QU, Sussex, England.
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Extract

While the general principles involved in the formulation of random walk and Brownian motion equations (whether the random changes are directly on the position of a particle or individual, or on the velocity) are well-known, there are various situations considered in the literature involving the assumption of a constant speed (in magnitude). Thus the derivation by Goldstein (1951) of a one-dimensional wave-like equation involved the tacit assumption Ut = ± a, where Ut is the vector velocity dRt/dt,Rt being the (column) position vector (Bartlett (1957)). Biological models may involve the assumption of individuals moving at constant speed (cf. Kendall (1974)). Finally, the derivation of Schrodinger-type equations from Brownian motion models has sometimes involved the assumption U′tUt = c2, where c is the velocity of light (Cane (1967), (1975)).

Type
Research Article
Information
Advances in Applied Probability , Volume 10 , Issue 4 , December 1978 , pp. 704 - 707
Copyright
Copyright © Applied Probability Trust 1978 

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References

Bartlett, M. S. (1957) Some problems associated with random velocity. Publ. Inst. Statist. Univ. Paris 6, 261–270.Google Scholar
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Cane, V. R. (1967) Random walks and physical processes. Bull. Internat. Statist. Inst. 42(1), 622–640.Google Scholar
Cane, V. R. (1975) Diffusion models with relativity effects. In Perspectives in Probability and Statistics, ed. Gani, J., distributed by Academic Press, London for the Applied Probability Trust, Sheffield, 263–273.Google Scholar
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Kendall, D. G. (1974) Pole-seeking Brownian motion and bird navigation. J.R. Statist. Soc. B 36, 365–417.Google Scholar