Local Asymptotic Laws for Brownian Motion

Abstract

Upper and lower functions are defined for the large values of $|X_d(t + u) - X_d(t - \nu)|$ as $(u + \nu) \downarrow 0$ where $X_d$ is a standard Brownian motion in $R^d$, and it is shown that the integral test for two-sided growth in $R^d$ is the same as that for one-sided growth in $R^{d+2}$. It is also shown that, for $d \geqq 4$, the lower asymptotic growth rate of $|X_d(t + u) - X_d(t - \nu)|$ for small $(u + \nu) = h$ is the same as the lower growth rate of $|X_{d-2}(t + h) - X_{d-2}(t)|$. Integral tests are also obtained for local asymptotic growth rates of the associated processes $P_d(a) = \inf_{t\geqq0} \{t: |X(t)| \geqq a\}$ and $M_d(t) = \sup_{0\leqq s\leqq t} |X_d(t)|$.