The most visited site of Brownian motion and simple random walk

Summary Let L(t, x) be the local time at x for Brownian motion and for each t, let , the absolute value of the most visited site for Brownian motion up to time t. In this paper we prove that ¯V(t) is transient and obtain upper and lower bounds for the rate of growth of ¯V(t). The main tools used are the Ray-Knight theorems and William's path decomposition of a diffusion. An invariance principle is used to get analogous results for simple random walks. We also obtain a law of the iterated logarithm for ¯V(t).This research was partially supported by NSF Grants MCS 83-00581 and MCS 83-03297