The Proportion of Brownian Sojourn Outside a Moving Boundary

Abstract

Let $B_t$ be a $d$-dimensional Brownian motion starting at zero and $h(t)$ a positive nondecreasing function of $t > 0$. It is shown that $\lim \sup_{s\downarrow 0}(1/s)\operatorname{meas} \{t \in (0, s\rbrack: |B_t| > h(t)\sqrt t\} = 1 - e^{-4(q-1)}$; where $q$ is defined as a simple functional of $h$ and, if we take $h_c(t) = c\sqrt{(2 \log \log(1/t))}$ for $h$ and if $0 < c \leqq 1, q = 1/c^2$. We also investigate $X^\ast = \lim \sup_{s\downarrow 0}(1/s)\operatorname{meas}\{t \in (0, s\rbrack: |B_t| < (\sqrt t)/h(t)\}$ and find upper and lower bounds of $X^\ast$, which indicate in particular that if $h = h_c(c > 0), X^\ast = p_c$ (say) is positive and less than one and tends to zero (one) as $c\uparrow \infty$ (respectively, $c \downarrow 0$). The problem for the case $s \rightarrow \infty$ is also treated.