تقرير
Remarks on the range and multiple range of random walk up to the time of exit
| العنوان: | Remarks on the range and multiple range of random walk up to the time of exit |
|---|---|
| المؤلفون: | Doehrman, Thomas, Sethuraman, Sunder, Venkataramani, Shankar C. |
| سنة النشر: | 2020 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Probability, 60G50, 60F05 |
| الوصف: | 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 $p\geq 1$ times, of simple random walk on ${\mathbb Z}^d$, for dimensions $d\geq 2$, up to time of exit from a domain $D_N$ of the form $D_N = ND$ where $D\subset {\mathbb R}^d$, as $N\uparrow\infty$. 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 $d\geq 3$, 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 $d\geq 2$, 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. Comment: 11 pages, 1 figure; updated intro and references |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2003.07960 |
| رقم الأكسشن: | edsarx.2003.07960 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |