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Capacity Estimates, Boundary Crossings and the Ornstein-Uhlenbeck Process in Wiener Space
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Endre Csáki, Hungarian Academy of Sciences
Davar Khoshnevisan, University of Utah
Zhan Shi, Université Paris VI
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Abstract
Let $T_1$ denote the first passage time to 1 of a standard
Brownian motion. It is well known that as $lambda$ goes to infinity,
$P{ T_1 > lambda }$ goes to zero at rate $c lambda^{-1/2}$, where $c$
equals $(2/ pi)^{1/2}$. The goal of this note is to establish a quantitative,
infinite dimensional version of this result. Namely, we will prove the existence
of positive and finite constants $K_1$ and $K_2$, such that for all $lambda>e^e$,
where `log' denotes the natural logarithm, and Cap is the Fukushima-Malliavin
capacity on the space of continuous functions.
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Full text: PDF
Pages: 103-109
Published on: November 20, 1999
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Bibliography
- Csáki, E., Khoshnevisan, D. and Shi, Z.,
Boundary crossings and the distribution function of the maximum of
Brownian sheet.
Preprint. Math Review Number not available
- Fukushima, M.,
Basic properties of Brownian motion and a capacity on the Wiener space.
J. Math. Soc. Japan 36, (1984) 161-176
Math. Review 85h:60114
- Malliavin, P.,
Stochastic calaculus of variation and hypoelliptic operators.
Proc. International Symp. Stoch. Diff. Eq. (Kyoto 1976), pp. 195-263. Wiley,
New York, 1978
Math. Review 81f:60083
- Uchiyama, K.,
Brownian first exit from and sojourn over one sided moving boundary and applications.
Z. Wahrsch. Verw. Gebiete, 54, pp. 75-116, 1980
Math. Review 82c:60143
- Williams, D.,
Appendix to P.-A. Meyer: Note sur les processus d'Ornstein-Uhlenbeck.
Sém. de Probab. XVI. Lecture Notes in Mathematics, 920, p. 133.
Springer, Berlin, 1982.
Math. Review 84i:60103
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Electronic Communications in Probability. ISSN: 1083-589X |
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