The “logarithmic scale” Minkowski dimension of the most visited sites of two-dimensional Brownian motion

Abstract

This paper studies the sets of visit times to points x on the plane, where a standard two-dimensional Brownian motion makes a substantial number of visits. For this purpose, we introduce the concept of the “logarithmic scale” Minkowski dimension as a tool for measuring the sets of visit times. We prove that, almost surely, there is a point x such that the “logarithmic scale” Minkowski dimension of the set of visit times to x is 1.