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On A Brownian Motion Problem of T. Salisbury

Published online by Cambridge University Press:  20 November 2018

Frank B. Knight
Frank B. Knight*
Affiliation:
Department of Mathematics University of Illinois 1409 West Green Street Urbana, Illinois 61801 U.S.A.
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Abstract

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Let B be a Brownian motion on R, B(0) = 0, and let f (t, x) be continuous. T. Salisbury conjectured that if the total variation of f (t, B(t)), 0 ≤ t ≤ 1, is finite P-a.s., then f does not depend on x. Here we prove that this is true if the expected total variation is finite.

Keywords

60J65
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 40 , Issue 1 , 01 March 1997 , pp. 67 - 71
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Chacon, R., Le Jan, Y., Perkins, E., and Taylor, S. J., Generalized arc length for Brownian motion and Lévy processes, Z. Wahrsch. Verw. Gebiete 57 (1981), 197–211.Google Scholar
2. Salisbury, T. S., An increasing diffusion. In: Seminar on Stochastic Processes, 1984, Birkhauser, Boston, (1986), 173–194.Google Scholar