Asymptotically one-dimensional diffusion on the Sierpiński gasket and multi-type branching processes with varying environment
Asymptotically one-dimensional processes on fractals were introduced by \textit{K. Hattori}, \textit{T. Hattori} and \textit{H. Watanabe} [Probab. Theory Relat. Fields 100, No. 1, 85-116 (1994; Zbl 0808.60067)]. For the Sierpiński gasket it was shown that there is a natural class of diffusions on this fractal, not equivalent in Euclidean space, which have rather different local properties to the Brownian motion. Due to homogenization effects they behave globally like the Brownian motion, yet locally they have a preferred direction of motion. \textit{B. M. Hambly} and \textit{T. Kumagai} [Potential Anal. 8, No. 4, 359-397 (1998; Zbl 0910.60062)] have given an alternative construction of the diffusion process using Dirichlet forms, extended the class of fractals considered to some nested fractals and obtained some new results. The aim of the reviewed paper is to investigate these problems using branching processes techniques. On the first step two side estimations for the left tail of the limiting distribution for a supercritical multi-type branching process with varying environment are obtained. Then the spectral dimension of the diffusion processes are calculated and short time heat kernel estimates in the Euclidean metric are obtained.
scientific article; zbMATH DE number 1784210