Scaling limit for Brownian motions on the $l$-level Sierpinski gaskets: The fractal to Euclidean crossover
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866909805753401344 |
|---|---|
| author | Croydon, David A. Hambly, Ben Kumagai, Takashi |
| author_facet | Croydon, David A. Hambly, Ben Kumagai, Takashi |
| contents | In two dimensions, the $l$-level Sierpinski gasket $\mathrm{SG}(l)$ is obtained by splitting an equilateral triangle into a collection of $l^2$ equilateral triangles of equal size and with the same total area, retaining only the $l(l+1)/2$ triangles with the same orientation as the original triangle, and then iterating this procedure indefinitely. We show that the canonical diffusions on the spaces $\mathrm{SG}(l)$, $l\geq2$, can be rescaled to yield Brownian motion on the initial triangle. Our argument also applies to the analogous higher-dimensional Sierpinski gaskets. Moreover, we prove a local central limit theorem for the associated transition densities. Key to this is the derivation of a Poincaré inequality, in the proof of which we exploit the Euclidean-type mixing that occurs between the bottlenecks present at each scale of the fractal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_20657 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Scaling limit for Brownian motions on the $l$-level Sierpinski gaskets: The fractal to Euclidean crossover Croydon, David A. Hambly, Ben Kumagai, Takashi Probability 28A80 (primary), 60J25, 60J35, 60K50 In two dimensions, the $l$-level Sierpinski gasket $\mathrm{SG}(l)$ is obtained by splitting an equilateral triangle into a collection of $l^2$ equilateral triangles of equal size and with the same total area, retaining only the $l(l+1)/2$ triangles with the same orientation as the original triangle, and then iterating this procedure indefinitely. We show that the canonical diffusions on the spaces $\mathrm{SG}(l)$, $l\geq2$, can be rescaled to yield Brownian motion on the initial triangle. Our argument also applies to the analogous higher-dimensional Sierpinski gaskets. Moreover, we prove a local central limit theorem for the associated transition densities. Key to this is the derivation of a Poincaré inequality, in the proof of which we exploit the Euclidean-type mixing that occurs between the bottlenecks present at each scale of the fractal. |
| title | Scaling limit for Brownian motions on the $l$-level Sierpinski gaskets: The fractal to Euclidean crossover |
| topic | Probability 28A80 (primary), 60J25, 60J35, 60K50 |
| url | https://arxiv.org/abs/2509.20657 |