Scaling limit for Brownian motions on the l-level Sierpinski gaskets: the fractal to Euclidean crossover

Journal article

In two dimensions, the l-level Sierpinski gasket 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 SG(l), l ≥ 2, 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.

Authors: Croydon, DA; Hambly, B; Kumagai, T

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Institute of Mathematical Statistics
• Journal: Electronic Journal of Probability
• Volume: 31
• Article number: 55
• Publication date: 2026-03-13
• Acceptance date: 2026-02-27
• DOI: 10.1214/26-EJP1514
• EISSN: 1083-6489
• Language: English
• Keywords: Sierpinski gasket; fractal; Brownian motion; scaling limit; transition density