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A correlated random walk model for two-dimensional diffusion

Published online by Cambridge University Press:  14 July 2016

Robin Henderson ,
Eric Renshaw  and
David Ford
Robin Henderson*
Affiliation:
British Nuclear Fuels Ltd
Eric Renshaw*
Affiliation:
University of Edinburgh
David Ford*
Affiliation:
Institute of Terrestrial Ecology
*
∗ Present address: Department of Statistics, University of Newcastle upon Tyne, Claremont Road, Newcastle upon Tyne NE1 7RU, U.K.
∗∗ Postal address: Department of Statistics, University of Edinburgh, James Clerk Maxwell Building, King's Buildings, Mayfield Rd, Edinburgh EH9 3JZ, U.K.
∗∗∗ Institute of Terrestrial Ecology, Bush Estate, Penicuik, Midlothian, EH26 0QB, U.K.
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Abstract

A two-dimensional lattice random walk is studied in which, at each stage, the direction of the next step is correlated to that of the previous step. An exact expression is obtained from the characteristic function for the dispersion matrix of the n-step transition probabilities, and the limiting diffusion equation is derived.

Keywords

LATTICEDISPERSIONCHARACTERISTIC FUNCTIONSPECTRAL DECOMPOSITION
Type
Research Papers
Information
Journal of Applied Probability , Volume 21 , Issue 2 , June 1984 , pp. 233 - 246
Copyright
Copyright © Applied Probability Trust 1984 

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References

Cane, V. R. (1975) Diffusion models with relativity effects. In Perspectives in Probability and Statistics: Papers in Honour of M. S. Bartlett on the Occasion of his Sixty-fifth Birthday, ed. Gani, J., distributed by Academic Press, London, for the Applied Probability Trust, Sheffield, 263–273.Google Scholar
Dvoretsky, A. and Erdös, P. (1950) Some problems on random walk in space. Proc. 2nd Berkeley Symp. Math. Statist. Prob., 353–367.Google Scholar
Feller, W. (1968) An Introduction to Probability Theory and its Applications, Vol. 1, 3rd edn. Wiley, New York.Google Scholar
Flory, P. J. (1962) Principles of Polymer Chemistry. Cornell University Press, Ithaca, New York.Google Scholar
Henderson, R. (1981) The Structural Root Systems of Sitka Spruce and Related Stochastic Processes. , University of Edinburgh.Google Scholar
Henderson, R. and Renshaw, E. (1980) Spatial stochastic models and computer simulation applied to the study of tree root systems. In COMPSTAT 1980, ed. Barritt, M. M. and Wishart, D., Physica Verlag, Vienna, 389–395.Google Scholar
Henderson, R., Renshaw, E. and Ford, E. D. (1983) A note on the recurrence of a correlated random walk, J. Appl. Prob. 20, 696–699.CrossRefGoogle Scholar
Holgate, P. (1966) Recurrence of sums of multiple Markov sequences. Israel J. Math. 4, 208–212.CrossRefGoogle Scholar
Klein, G. (1952) A generalisation of the random walk problem. Proc. R. Soc. Edinburgh A63, 268–279.Google Scholar
Montroll, E. W. (1964) Random walks on lattices. Proc. Symp. Appl. Math. Amer. Math. Soc. 16, 193–220.CrossRefGoogle Scholar
Pielou, E. C. (1977) Mathematical Ecology, 2nd edn. Wiley, New York.Google Scholar
Renshaw, E. and Henderson, R. (1981) The correlated random walk. J. Appl. Prob. 18, 403–414.Google Scholar
Skellam, J. G. (1973) The formulation and interpretation of mathematical models of diffusionary processes in population biology. In The Mathematical Theory of the Dynamics of Biological Populations, ed. Bartlett, M. S. and Hiorns, R. W., Academic Press, London, 63–85.Google Scholar