Some New Classes of Exceptional Times of Linear Brownian Motion

Abstract

We study certain classes of exceptional times of a linear Brownian motion $(B_t, t \geq 0)$. In particular, we consider the set $K^-$ of all instants $t \in \lbrack 0, 1\rbrack$ such that the value $B_t$ of the Brownian motion at time $t$ is greater than its mean value over all intervals $\lbrack s,t\rbrack, s < t$. We also study the subset $K$ of $K^-$ of all instants $t$ such that in addition $B_t$ is greater than the mean value of $B$ over the intervals $\lbrack t,s\rbrack, t < s \leq 1$. We compute the Hausdorff dimension of $K^-, K$ and some other related sets of exceptional times. These results are closely related to a recent work of Sinai motivated by the analysis of solutions to the Burgers equation with random initial data. The proofs involve studying suitable approximations of the sets $K^-$ and $K$, and deriving precise estimates for the probability that a given time $t$ belongs to these approximations. A delicate zero-one law argument is also needed to prove that the lower bound on the dimension of $K$ holds with probability 1.