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 df = log N / log r. This result demonstrates that fractal geometry scales the limiting distributions and dictates the stochastic regime.
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BibTeX
@misc{takahashi2026geometric,
title = {{Geometric influence on the limiting behavior of diffusion processes in one-sided Brownian environments on disconnected fractal sets}},
author = {Hiroshi Takahashi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08195},
eprint = {2609.08195},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08195},
}
RIS
TY - UNPB
TI - Geometric influence on the limiting behavior of diffusion processes in one-sided Brownian environments on disconnected fractal sets
AU - Hiroshi Takahashi
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08195
UR - https://arxiv.org/abs/2609.08195
AB - 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.
ER -