TY - JOUR
T1 - REMARKS ON THE RANGE AND MULTIPLE RANGE OF A RANDOM WALK UP TO THE TIME OF EXIT
AU - Doehrman, Thomas
AU - Sethuraman, Sunder
AU - Venkataramani, Shankar C.
N1 - Publisher Copyright:
© 2021 Rocky Mountain Mathematics Consortium. All rights reserved.
PY - 2021/10
Y1 - 2021/10
N2 - 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.
AB - 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.
KW - And phrases: random walk
KW - Brownian motion
KW - Constrained
KW - Exit
KW - Multiple
KW - Polyharmonic
KW - Range
KW - Time
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U2 - 10.1216/rmj.2021.51.1603
DO - 10.1216/rmj.2021.51.1603
M3 - Article
AN - SCOPUS:85126334066
SN - 0035-7596
VL - 51
SP - 1603
EP - 1614
JO - Rocky Mountain Journal of Mathematics
JF - Rocky Mountain Journal of Mathematics
IS - 5
ER -