论文标题
关于范围和多个随机步行到退出时间的备注
Remarks on the range and multiple range of random walk up to the time of exit
论文作者
论文摘要
我们考虑范围的缩放行为和$ p $ - 万月范围，即访问的积分数和访问的点数$ p \ geq 1 $ times，在$ {\ mathbb z}^d $上进行简单的随机步行，对于尺寸$ d \ geq 2 $，从域名$ d_n $ d_ $ d_ d_ d_ d_ d_ d_ d_ r}^d $，as $ n \ uparrow \ infty $。最近的论文讨论了该范围和相关统计数据与高斯自由领域的连接，特别是确定该范围的分布缩放限制（如果是$ d $是$ d \ geq 3 $中的立方体）与布朗尼运动的退出时间成比例。本说明的目的是给出一个简洁，不同的论点，即在$ d \ geq 2 $中的一般环境中，缩放范围和多个范围都微弱地融合到$ d $的布朗运动的比例退出时间，相应的限制矩是“多旋转量”，求解了Poisson方程的层次。
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.
