发表机构
Keio University; Waseda University（庆应义塾大学; 早稻田大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究二维随机游走碰撞测度的对数尺度极限，证明其依分布收敛到由泊松点过程表示的随机测度，并通过簇分解揭示碰撞结构。
AI 中文摘要
我们研究了在 $\mathbb Z^2$ 上两个独立同分布的离散时间随机游走的碰撞测度的尺度极限，该测度记录它们的碰撞位置和时间。这是一个碰撞的临界状态：游走无限次碰撞，而极限布朗运动在正时间不会碰撞。因此，现有的关于碰撞测度收敛的一般结果不适用。假设跳跃分布具有零均值和有限二阶矩，我们证明在对数尺度下，碰撞测度依分布收敛到一个非平凡的随机测度。我们还给出了极限测度在泊松点过程方面的显式表示。为了证明收敛性，我们引入了碰撞测度拉普拉斯变换的一种新的簇分解，揭示了碰撞的簇结构。
英文摘要
We study the scaling limit of the collision measure of two i.i.d. discrete-time random walks on $\mathbb Z^2$, which records their collision sites and times. This is a critical regime for collisions: the walks collide infinitely often, whereas the limiting Brownian motions do not collide at positive times. Hence, existing general results on the convergence of collision measures do not apply. Assuming that the jump distribution has mean zero and finite second moment, we prove that, under a logarithmic scaling, the collision measure converges in distribution to a non-trivial random measure. We also give an explicit representation of the limiting measure in terms of a Poisson point process. To prove convergence, we introduce a new cluster decomposition of the Laplace transform of the collision measure, revealing the cluster structure of collisions.
Comments33 pages, no figure
