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One-dimensional classical scattering processes and the diffusion limit

Published online by Cambridge University Press:  01 July 2016

Wilfrid S. Kendall  and
Mark Westcott
Wilfrid S. Kendall*
Affiliation:
University of Strathclyde
Mark Westcott*
Affiliation:
CSIRO Division of Mathematics and Statistics
*
∗Postal address: Department of Mathematics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow Gl 1XH, UK.
∗∗ Postal address: Division of Mathematics and Statistics, CSIRO PO Box 1965, Canberra City ACT 2601, Australia.
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Abstract

The method of Skorokhod embedding, that is to say of coupling to Brownian motion, is used to establish the asymptotic behaviour of one-dimensional scattering processes.

Keywords

RAYLEIGH RANDOM FLIGHTSCONGLOMERATIONBROWNIAN MOTIONDIFFUSION CONSTANTSKOROKHOD EMBEDDINGCOUPLINGPROBABILISTIC AVERAGINGLOCAL TIME
Type
Research Article
Information
Advances in Applied Probability , Volume 19 , Issue 1 , March 1987 , pp. 81 - 105
Copyright
Copyright © Applied Probability Trust 1987 

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References

Bartlett, M. S. (1957) Some problems associated with random velocity. Publ. Inst. Statist. Univ. Paris 6 (4), 261–270.Google Scholar
Bingham, N. H. and Goldie, C. M. (1982) Probabilistic and deterministic averaging. Trans. Amer. Math. Soc. 269, 453–480.Google Scholar
Cane, V. (1967) Random walks and physical processes. Bull. Internat. Stat. Inst. 42, 622–640.Google Scholar
Cane, V. (1975) Diffusion models with relativity effects, in Perspectives in Probability and Statistics, ed. Gani, J., Applied Probability Trust, Sheffield, 263–273.Google Scholar
Chow, Y. S. and Teicher, H. (1978) Probability Theory. Springer-Verlag, Berlin.Google Scholar
Gates, D. J. and Westcott, M. (1982) Diffusion laws for random walks on various heterogeneous lattices. J. Phys. A 15, L267–L271.Google Scholar
Gillis, J. (1955) Correlated random walk. Proc. Camb. Phil. Soc. 51, 639–651.Google Scholar
Gorostiza, L. (1972) An invariance principle for a class of d-dimensional polygonal random functions. Trans. Amer. Math. Soc. 177, 413–445.Google Scholar
Henderson, R., Renshaw, E. and Ford, D. (1983) A note on the recurrence of a correlated random walk. J. Appl. Prob. 20, 696–699.Google Scholar
Henderson, R., Renshaw, E. and Ford, D. (1984) A correlated random walk model for two-dimensional diffusion. J. Appl. Prob. 21, 233–246.CrossRefGoogle Scholar
Kalikow, S. (1981) Generalized random walk in a random environment. Ann. Prob. 9, 753–768.Google Scholar
Keilson, J. and Wellner, J. A. (1978) Oscillating Brownian motion. J. Appl. Prob. 15, 300–310.CrossRefGoogle Scholar
Kesten, H., Koslov, M. and Spitzer, F. (1975) A limit theorem for random walk in a random environment. Compositio Math. 30, 145–168.Google Scholar
Lawler, G. F. (1982) Weak convergence of a random walk in a random environment Commun. Math. Phys. 87, 81–87.Google Scholar
Papanicolauou, C. and Varadhan, S. R. S. (1982) Diffusions with random coefficients, in Essays in Honour of C. R. Rao, North-Holland, Amsterdam, 547–552.Google Scholar
Rayleigh, Lord (1919) On the problem of random vibrations, and of random flights in one, two, or three dimensions. Phil. Mag. (6th Ser.) 37, 321–347.Google Scholar
Redheffer, R. (1960) The Mycielski-Paszkowski diffusion problem. J. Math. Mech. 9, 607–621.Google Scholar
Renshaw, E. and Henderson, R. (1981) The correlated random walk. J. Appl. Prob. 18, 403–414.CrossRefGoogle Scholar
Revesz, P. (1978) Laws of Large Numbers, Academic Press, New York.Google Scholar
Shuler, K. E. and Mohanty, U. (1981) Random walk properties from lattice bond enumeration: Anisotropic diffusion in lattices with periodic and randomly distributed scatterers. Proc. Nat. Acad. Sci. USA (Phys. Sci) 78, 6576–6578.Google Scholar
Skorokhod, A. V. and Slobodenyuk, N. P. (1970) Limit Theorems for Random Walks (in Russian). Izd-vo “Naukova Dymka”, Kiev.Google Scholar
Williams, D. (1979) Diffusions, Markov Processes, and Martingales, Vol 1: Foundations. Wiley, New York.Google Scholar