The logarithmic scaleMinkowski dimension of the most visited sites of two-dimensional Brownian motion

This paper studies the sets of visit times to pointsx 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 scaleMinkowski 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 scaleMinkowski dimension of the set of visit times to x is 1.

Automated summary

This paper studies the sets of visit times to points on the plane by a standard two-dimensional Brownian motion. The concept of logarithmic scale Minkowski dimension is introduced as a tool for measuring these sets. It is proved that almost surely there exists a point x such that the logarithmic scale Minkowski dimension of the set of visit times to x is 1.