The most visited site of Brownian motion and simple random walk

  • Richard F. Bass
  • Philip S. Griffin


Let L(t, x) be the local time at x for Brownian motion and for each t, let \(\bar V(t) = \inf \{ x\underline{\underline > } 0;L(t,x) \vee L(t, - x) = \mathop {\sup }\limits_y L(t,y)\} \), 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).


Lower Bound Stochastic Process Brownian Motion Random Walk Probability Theory 


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Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Richard F. Bass
    • 1
  • Philip S. Griffin
    • 1
  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

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