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Homogenization of anisotropic diffusion on pre-Sierpiński carpets

  • Shiping Cao
  • Zhen-Qing Chen
  • Hua Qiu
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Abstract

We investigate the restoration of isotropy for anisotropic diffusions on pre-Sierpiński carpets in the plane, a problem previously studied by Barlow, Hattori, Hattori and Watanabe \cite{BHHW} in which they obtained a weak homogenization property for the ratio of effective resistances of the anisotropic diffusions. We establish the weak convergence of these anisotropic diffusions to Brownian motion on the Sierpiński carpet. As a consequence, we give an affirmative answer to the strong homogenization conjecture raised in \cite[p.3]{BHHW}.

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Cite

BibTeXRIS
BibTeX
@misc{cao2026homogenization,
  title = {{Homogenization of anisotropic diffusion on pre-Sierpiński carpets}},
  author = {Shiping Cao and Zhen-Qing Chen and Hua Qiu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04936},
  eprint = {2609.04936},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04936},
}
RIS
TY  - UNPB
TI  - Homogenization of anisotropic diffusion on pre-Sierpiński carpets
AU  - Shiping Cao
AU  - Zhen-Qing Chen
AU  - Hua Qiu
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.04936
UR  - https://arxiv.org/abs/2609.04936
AB  - We investigate the restoration of isotropy for anisotropic diffusions on pre-Sierpiński carpets in the plane, a problem previously studied by Barlow, Hattori, Hattori and Watanabe \cite{BHHW} in which they obtained a weak homogenization property for the ratio of effective resistances of the anisotropic diffusions. We establish the weak convergence of these anisotropic diffusions to Brownian motion on the Sierpiński carpet. As a consequence, we give an affirmative answer to the strong homogenization conjecture raised in \cite[p.3]{BHHW}.
ER  -