REMARKS ON THE RANGE AND MULTIPLE RANGE OF A RANDOM WALK UP TO THE TIME OF EXIT

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Abstract

We consider the scaling behavior of the range and p-multiple range, that is the number of points visited and the number of points visited exactly p ≥ 1 times, of a simple random walk on Zd, for dimensions d ≥ 2, up to time of exit from a domain DN of the form DN = ND, where D ⊂ Rd, as N ↑ ∞. Recent papers have discussed connections of the range and related statistics with the Gaussian free field, identifying in particular that the distributional scaling limit for the range, in the case D is a cube in d ≥ 3, is proportional to the exit time of Brownian motion. The purpose of this note is to give a concise, different argument that the scaled range and multiple range, in a general setting in d ≥ 2, both weakly converge to proportional exit times of Brownian motion from D, and that the corresponding limit moments are “polyharmonic”, solving a hierarchy of Poisson equations.

Original languageEnglish (US)
Pages (from-to)1603-1614
Number of pages12
JournalRocky Mountain Journal of Mathematics
Volume51
Issue number5
DOIs
StatePublished - Oct 2021

Keywords

  • And phrases: random walk
  • Brownian motion
  • Constrained
  • Exit
  • Multiple
  • Polyharmonic
  • Range
  • Time

ASJC Scopus subject areas

  • General Mathematics

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