Summary
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).
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This research was partially supported by NSF Grants MCS 83-00581 and MCS 83-03297
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Bass, R.F., Griffin, P.S. The most visited site of Brownian motion and simple random walk. Z. Wahrscheinlichkeitstheorie verw Gebiete 70, 417–436 (1985). https://doi.org/10.1007/BF00534873
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DOI: https://doi.org/10.1007/BF00534873