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The most visited sites of symmetric stable processes
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  • Published: March 2000

The most visited sites of symmetric stable processes

  • Richard F. Bass1,
  • Nathalie Eisenbaum2 &
  • Zhan Shi2 

Probability Theory and Related Fields volume 116, pages 391–404 (2000)Cite this article

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  • 19 Citations

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Abstract.

Let X be a symmetric stable process of index α∈ (1,2] and let L x t denote the local time at time t and position x. Let V(t) be such that L t V(t) = sup x∈ ℝ L t x. We call V(t) the most visited site of X up to time t. We prove the transience of V, that is, lim t →∞ |V(t)| = ∞ almost surely. An estimate is given concerning the rate of escape of V. The result extends a well-known theorem of Bass and Griffin for Brownian motion. Our approach is based upon an extension of the Ray–Knight theorem for symmetric Markov processes, and relates stable local times to fractional Brownian motion and further to the winding problem for planar Brownian motion.

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Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009 USA. e-mail: bass@math.uconn.edu, , , , , , US

    Richard F. Bass

  2. Laboratoire de Probabilités, Université Paris VI, 4 Place Jussieu, 75252 Paris Cedex 05, France. e-mail: nae@ccr.jussieu.fr; e-mail: zhan@proba.jussieu.fr, , , , , , FR

    Nathalie Eisenbaum & Zhan Shi

Authors
  1. Richard F. Bass
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  2. Nathalie Eisenbaum
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  3. Zhan Shi
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Received: 14 October 1998 / Revised version: 8 June 1999 / Published online: 7 February 2000

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Bass, R., Eisenbaum, N. & Shi, Z. The most visited sites of symmetric stable processes. Probab Theory Relat Fields 116, 391–404 (2000). https://doi.org/10.1007/s004400050255

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  • Issue Date: March 2000

  • DOI: https://doi.org/10.1007/s004400050255

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Keywords

  • Brownian Motion
  • Markov Process
  • Local Time
  • Stable Process
  • Fractional Brownian Motion
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