Capacity Estimates, Boundary Crossings and the Ornstein-Uhlenbeck Process in Wiener Space
Endre Csáki, Hungarian Academy of Sciences
Davar Khoshnevisan, University of Utah
Zhan Shi, Université Paris VI
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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