Where Did the Brownian Particle Go?
Yuval Peres (University of California, Berkeley)
Jim Pitman (University of California, Berkeley)
Marc Yor (Université Pierre et Marie Curie)
Abstract
Consider the radial projection onto the unit sphere of the path a $d$-dimensional Brownian motion $W$, started at the center of the sphere and run for unit time. Given the occupation measure $\mu$ of this projected path, what can be said about the terminal point $W(1)$, or about the range of the original path? In any dimension, for each Borel set $A$ in $S^{d-1}$, the conditional probability that the projection of $W(1)$ is in $A$ given $\mu(A)$ is just $\mu(A)$. Nevertheless, in dimension $d \ge 3$, both the range and the terminal point of $W$ can be recovered with probability 1 from $\mu$. In particular, for $d \ge 3$ the conditional law of the projection of $W(1)$ given $\mu$ is not $\mu$. In dimension 2 we conjecture that the projection of $W(1)$ cannot be recovered almost surely from $\mu$, and show that the conditional law of the projection of $W(1)$ given $\mu$ is not $mu$.
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Pages: 1-22
Publication Date: January 10, 2001
DOI: 10.1214/EJP.v6-83
References
- D. J. Aldous, Brownian excursion conditioned on its local time, Electron. Comm. Probab. 3 (1998), 79-90. Math. Reviews link 99m:60115
- M. Barlow, J. Pitman and M. Yor, On Walsh's Brownian motions, Séminaire de Probabilités XXIII, (1998) 275-293. Math. Reviews link 91a:60204
- R. F. Bass and D. Khoshnevisan. Local times on curves and uniform invariance principles, Probab. Th. Rel. Fields 92, (1992), 465-492, Math. Reviews link 93e:60161
- N. H. Bingham and R. A. Doney, On higher-dimensional analogues of the arc-sine law, Journal of Applied Probability 25, (1988) 120-131, Math. Reviews link 89g:60249
- K. Burdzy. Cut points on Brownian paths. Ann. Probab. 17, (1989) 1012-1036, Math. Reviews link 90m:60091
- K. Burdzy, Labyrinth dimension of Brownian trace. Probab. Math. Statist. 15, (1995) 165-193, Math. Reviews link 97j:60146
- K. Burdzy and G. Lawler. Nonintersection exponents for Brownian paths. II. Estimates and applications to a random fractal. Ann. Probab. 18, (1990) 981-1009, Math. Reviews link 91g:60097
- P. Carmona, F. Petit and M. Yor, Some extensions of the arc sine law as partial consequences of the scaling property of Brownian motion. Probab. Th. Rel. Fields 100, (1994) 1-29, Math. Reviews link 95f:60089
- P. Carmona, F. Petit, and M. Yor. Beta variables as times spent in [0,infty[ by certain perturbed Brownian motions. J. London Math. Soc. (2), 58(1), (1998), 239-256, Math. Reviews link 2000b:60186
- L. Chaumont and R. Doney. A stochastic calculus approach to doubly perturbed Brownian motions. Preprint, Univ. Paris VI, 1996. Math. Reviews number not available
- A. Dembo, Y. Peres, J. Rosen and O. Zeitouni. Thick points for spatial Brownian motion: multifractal analysis of occupation measure, Ann. Probab. 28, (2000) 1-35, Math. Reviews link 1 755 996
- A. Dembo, Y. Peres, J. Rosen and O. Zeitouni. Thick points for planar Brownian motion and the Erdos-Taylor conjecture on random walk, To appear in Acta Math, 2000. Math. reviews number not available.
- R. Durrett. Probability: Theory and Examples, 2nd edition. Duxbury Press: Belmont, CA, 1996. Math. Reviews link 98m:60001
- T.S. Ferguson. Prior distributions on spaces of probability measures. Ann. Statist., (1974) 615-629, Math. Reviews link 55 #11479
- R.K. Getoor and M.J. Sharpe. On the arc-sine laws for Lévy processes. J. Appl. Probab. 31, (1994) 76 - 89, Math. Reviews link 94k:60111
- T. Ignatov. On a constant arising in the theory of symmetric groups and on Poisson-Dirichlet measures . Theory Probab. Appl. 27, 136-147, 1982. Math. Reviews link 83i:10072
- K. Itô and H. P. McKean, Diffusion Processes and Their Sample Paths, Second printing. Springer-Verlag (1974) Math. Reviews link 49:9963
- J. F. C. Kingman. Random discrete distributions. J. Roy. Statist. Soc. B 37, (1975) 1-22, Math. Reviews link 51 #4505
- F. B. Knight. On the upcrossing chains of stopped Brownian motion . In J. Azema, M. Emery, M. Ledoux, and M. Yor, editors, Séminaire de Probabilités XXXII , pages 343-375. Springer, 1998. Lecture Notes in Math. 1686. Math. Reviews link 2000j:60099
- J. Lamperti. Semi-stable stochastic processes. Trans. Amer. Math. Soc. , 104 (1962) 62-78. Math. Reviews link 25 #1575
- J. Lamperti. Semi-stable Markov processes I . Z. Wahrsch. Verw. Gebiete , 22, (1972) 205-225. Math. Reviews link 46 #6478
- P. Lévy. Sur certains processus stochastiques homogènes. Compositio Math. , 7, (1939), 283-339. Math. Reviews link 1,150a
- E.A. Pecherskii and B.A. Rogozin. On joint distributions of random variables associated with fluctuations of a process with independent increments. Theory Prob. Appl. , 14, (1969) 410-423. Math. Reviews link 41 #4634
- E. A. Perkins and S. J. Taylor. Uniform measure results for the image of subsets under Brownian motion. Probab. Theory Related Fields 76, (1987) 257-289. Math. Reviews link 88m:60122
- M. Perman, J. Pitman, and M. Yor. Size-biased sampling of Poisson point processes and excursions. Probab. Th. Rel. Fields , 92, (1992) 21-39. Math. Reviews link 93d:60088
- M. Perman and W. Werner. Perturbed Brownian motions . Probab. Th. Rel. Fields , 108 (1997), 357-383. Math. Reviews link 98i:60081
- F. Petit. Quelques extensions de la loi de l'arc sinus. C.R. Acad. Sc. Paris, Serie I , 315, (1992) 855-858, 1992. 93g:60176
- J. Pitman and M. Yor. Arcsine laws and interval partitions derived from a stable subordinator. Proc. London Math. Soc. (3) , 65 (1992), 326-356. 93e:60152
- J. Pitman and M. Yor. Quelques identites en loi pour les processus de Bessel . In Hommage a P.A. Meyer et J. Neveu , Asterisque, pages 249-276. Soc. Math. de France, 1996. 98c:60106
- J. Pitman and M. Yor. Random discrete distributions derived from self-similar random s>ets . Electronic J. Probability , 1:Paper 4, (1996) 1-28. 98i:60010
- P.S. Puri and H. Rubin. A characterization based on the absolute difference of two i.i.d. random variables. Ann. Math. Stat. , 41, (1970) 2113-2122. 45 #2836
- D. Revuz and M. Yor. Continuous martingales and Brownian motion. Springer, Berlin-Heidelberg, 1994. 2nd edition. 95h:60072
- J. Sethuraman. A constructive definition of Dirichlet priors . Statistica Sinica , 4, (1994) 639-650. 95m:62058
- M. S. Taqqu. A bibliographical guide to self-similar processes and long-range dependence . In Dependence in Probab. and Stat.: A Survey of Recent Results; Ernst Eberlein, Murad S. Taqqu (Ed.) (1986) 137-162. Birkhauser (Basel, Boston) , 88g:60091
- J. Walsh. A diffusion with a discontinuous local time. In Temps Locaux , volume 52-53 of Asterisque , pages 37-45. Soc. Math. de France, 1978. 81b:60042
- J. Warren and M. Yor. The Brownian burglar: conditioning Brownian motion by its local time process . In J. Azema, M. Emery, , M. Ledoux, and M. Yor, editors, Séminaire de Probabilités XXXII , pages 328-342. Springer, 1998. Lecture Notes in Math. 1686. 99k:60208
- M. Yor. Some Aspects of Brownian Motion . Lectures in Math., ETH Zurich. Birkhauser, 1992. Part I: Some Special Functionals, 93i:60155

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