Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Non-independence of excursions of the Brownian sheet and of additive Brownian motion

Authors: Robert C. Dalang and T. Mountford
Journal: Trans. Amer. Math. Soc. 355 (2003), 967-985
MSC (2000): Primary 60G60; Secondary 60G17, 60G15
Published electronically: November 1, 2002
MathSciNet review: 1938741
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Abstract: A classical and important property of Brownian motion is that given its zero set, distinct excursions away from zero are independent. In this paper, we examine the analogous question for the Brownian sheet, and also for additive Brownian motion. Our main result is that given the level set of the Brownian sheet at level zero, distinct excursions of the sheet away from zero are not independent. In fact, given the zero set of the Brownian sheet in the entire non-negative quadrant, and the sign of all but a finite number of excursions away from zero, the signs of the remaining excursions are determined. For additive Brownian motion, we prove the following definitive result: given the zero set of additive Brownian motion and the sign of a single excursion, the signs of all other excursions are determined. In an appendix by John B. Walsh, it is shown that given the absolute value of the sheet in the entire quadrant and, in addition, the sign of the sheet at a fixed, non-random time point, then the whole sheet can be recovered.

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Additional Information

Robert C. Dalang
Affiliation: Département de Mathématiques, Ecole Polytechnique Fédérale, 1015 Lausanne, Switzerland

T. Mountford
Affiliation: Département de Mathématiques, Ecole Polytechnique Fédérale, 1015 Lausanne, Switzerland and Department of Mathematics, University of California, Los Angeles, California 90024

Keywords: Brownian sheet, excursions, level sets, additive Brownian motion
Received by editor(s): November 5, 2001
Received by editor(s) in revised form: June 21, 2002
Published electronically: November 1, 2002
Additional Notes: The research of the first author is partially supported by the Swiss National Foundation for Scientific Research.
The research of the second author was partially supported by NSF grant DMS-00-71471 and by the BRIMS Institute (Bristol).
Article copyright: © Copyright 2002 American Mathematical Society