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One-dimensional Brownian motion and the three-dimensional Bessel process

Published online by Cambridge University Press:  01 July 2016

James W. Pitman
James W. Pitman*
Affiliation:
University of Copenhagen∗
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Abstract

A simple path transformation is described which connects one-dimensional Brownian motion with the radial part of three-dimensional Brownian motion. This provides simple proofs of various path decompositions for these processes described by David Williams.

Keywords

BROWNIAN MOTIONBESSEL PROCESSPATH DECOMPOSITIONWEAK CONVERGENCE OF RANDOM WALKS TO DIFFUSIONS
Type
Research Article
Information
Advances in Applied Probability , Volume 7 , Issue 3 , September 1975 , pp. 511 - 526
Copyright
Copyright © Applied Probability Trust 1975 

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References

[1] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York and London.Google Scholar
[2] Breiman, L. (1968) Probability. Addison Wesley, Reading, Mass. and London, Ontario.Google Scholar
[3] Doob, J. L. (1957) Conditional Brownian motion and the boundary limits of harmonic functions. Bull. Soc. Math. France 85, 431–458.Google Scholar
[4] Dynkin, E. B. (1965) Markov Processes (English translation in two volumes). Springer, Berlin, Heidelberg, New York.Google Scholar
[5] Feller, W. (1968) An Introduction to Probability Theory and Its Applications, Volume 1, 3rd Ed. Wiley, New York and London.Google Scholar
[6] Feller, W. (1971) An Introduction to Probability Theory and Its Applications, Volume 2. 2nd Ed. Wiley, New York and London.Google Scholar
[7] Freedman, D. (1971) Markov Chains. Holden-Day, San Francisco, Cambridge, London, Amsterdam.Google Scholar
[8] Feller, W. (1971) Brownian Motion and Diffusion. Holden-Day, San Francisco, Cambridge, London, Amsterdam.Google Scholar
[9] Jacobsen, M. (1974) Splitting times for Markov processes and a generalised Markov property for diffusions. Z. Wahrscheinlichkeitsth. 30, 27–43.Google Scholar
[10] Knight, F. B. (1969) Brownian local times and taboo processes. Trans. Amer. Math. Soc. 143, 173–185.Google Scholar
[11] Itô, K. and McKean, H. P. (1965) Diffusion Processes and Their Sample Paths. Springer, Berlin, Heidelberg, New York.Google Scholar
[12] Lamperti, J. (1962) A new class of probability limit theorems. J. Math. Mech. 11, 749–772.Google Scholar
[13] Lévy, P. (1965) Processus Stochastique et Mouvement Brownien. Gauthier-Villars, Paris.Google Scholar
[14] McKean, H. P. (1963) Excursions of a non-singular diffusion. Z. Wahrscheinlichkeitsth. 1, 230–239.CrossRefGoogle Scholar
[15] Pitman, J. W. (1974) One-dimensional Brownian motion and the three-dimensional Bessel process. Proc. 3rd Conference on Stochastic Processes and Their Applications. Adv. Appl. Prob. 6, 223–224.Google Scholar
[16] Whitt, W. (1970) Weak convergence of probability measures on the function space C[0, ∞]. Ann. Math. Statist. 41, 939–944.Google Scholar
[17] Williams, D. (1970) Decomposing the Brownian path. Bull. Amer. Math. Soc. 76, 871–873.Google Scholar
[18] Williams, D. (1974) Path decomposition and continuity of local time for one-dimensional diffusions, I. Proc. London Math. Soc. (3) 28, 738–768.Google Scholar