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 - Funding Information:
We would like to thank Davar Khoshnevisan for useful conversations, and the referee for helpful suggestions with respect to the moment bound estimates in Section 3. S. S. was partially supported by ARO W911NF-18-1-0311 and a Simons Foundation Sabbatical grant. S. C. V. was partially supported by the Simons Foundation award 560103 and by the NSF award DMR-1923922.
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 -