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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Numerical bounds for critical exponents of crossing Brownian motion


Author: Mario V. Wüthrich
Journal: Proc. Amer. Math. Soc. 130 (2002), 217-225
MSC (2000): Primary 60K35, 82D30
Published electronically: May 22, 2001
MathSciNet review: 1855632
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Abstract:

We consider $d$-dimensional crossing Brownian motion in a truncated Poissonian potential conditioned to reach a fixed hyperplane at distance $L$from the starting point. The transverse fluctuation of the path is expected to be of order $L^\xi$. We prove that for $d\ge2$: $\xi \le 3/4$. As a second critical exponent we introduce $\chi^{(2)}$, which describes the fluctuations of naturally defined distance functions for crossing Brownian motion. The numerical bound we obtain is an improvement of Corollary 3.1 in Scaling identity for crossing Brownian motion in a Poissonian potential (Probab. Theory Related Fields 112 (1998), 299-319), resulting in $\chi^{(2)} \ge 1/5$ if $d=2$ and if the killing rate $\lambda$ is strictly positive ($\lambda>0$).


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Additional Information

Mario V. Wüthrich
Affiliation: Department of Mathematics, University of Nijmegen, Toernooiveld 1, NL-6525 ED Nijmegen, The Netherlands
Address at time of publication: Winterthur Insurance, Roemerstrasse 17, P.O. Box 357, CH-8401 Winterthur, Switzerland
Email: mario.wuethrich@winterthur.ch

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06017-8
PII: S 0002-9939(01)06017-8
Keywords: Brownian motion, Poissonian potential, fluctutation, critical exponents, superdiffusivity
Received by editor(s): September 1, 1999
Received by editor(s) in revised form: May 24, 2000
Published electronically: May 22, 2001
Communicated by: Claudia M. Neuhauser
Article copyright: © Copyright 2001 American Mathematical Society