Capacity Estimates, Boundary Crossings and the Ornstein-Uhlenbeck Process in Wiener Space

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 $\lambda>e^e$, $$K_1 \lambda^{-1/2} \leq \text{Cap} \{ T_1 > \lambda\} \leq K_2 \lambda^{-1/2} \log^3(\lambda) \cdot \log\log(\lambda),$$ where `$\log$' denotes the natural logarithm, and $\text{Cap}$ is the Fukushima-Malliavin capacity on the space of continuous functions.