发表机构
Keio University（庆应义塾大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究单侧布朗环境中不连通分形集上的扩散过程，发现其扩散或捕获行为由几何参数r和N决定，概率由豪斯多夫维数刻画。
AI 中文摘要
我们研究单侧布朗环境中不连通分形集上的扩散过程。在实直线上，已确立该过程要么表现出扩散，要么表现出捕获，每种情况发生的概率均为 $1/2$。在本研究中，我们证明对于分形集，这种行为由定义分形结构的几何参数 $r$（相似比的倒数）和 $N$（压缩映射的个数）所控制。出现了两种不同的状态：在无环境一侧为扩散状态，在受环境影响的一侧为局域化状态。这两种状态之间的转变由随机环境首先达到阈值 $\log r$ 还是 $-\log N$ 决定。因此，扩散与捕获的概率由豪斯多夫维数 $d_f = \log N / \log r$ 显式刻画。这一结果表明，分形几何对极限分布进行缩放，并决定随机状态。
英文摘要
We investigate diffusion processes on disconnected fractal sets in one-sided Brownian environments. On the real line, it is established that the process exhibits either diffusion or trapping, each occurring with probability $1/2$. In this study, we demonstrate that for fractal sets, this behavior is governed by the geometric parameters $r$ (the reciprocal of the similitude ratio) and $N$ (the number of contraction mappings), which define the fractal structure. Two distinct regimes emerge: a diffusive regime on the environment-free side and a localization regime on the side influenced by the environment. The transition between these regimes is determined by whether the random environment first hits the threshold $\log r$ or $-\log N$. Consequently, the probability of diffusion versus trapping is explicitly characterized by the Hausdorff dimension $d_f = \log N / \log r$. This result demonstrates that fractal geometry scales the limiting distributions and dictates the stochastic regime.
