Geometric influence on the limiting behavior of diffusion processes in one-sided Brownian environments on disconnected fractal sets

Abstract

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.