Random Walks on Sierpiński Graphs: Hyperbolicity and Stochastic Homogenization
Vadim A. Kaimanovich · Birkhäuser Basel eBooks · 2003
We introduce two new techniques to the analysis on fractals. One is based on the presentation of the fractal as the boundary of a countable Gromov hyperbolic graph, whereas the other one consists in taking all possible “backward” extensions of the above hyperbolic graph and considering them as the classes of a discrete equivalence relation on an appropriate compact space. Illustrating these techniques on the example of the Sierpiński gasket (the associated hyperbolic graph is called the Sierpiński graph), we show that the Sierpiński gasket can be identified with the Martin and the Poisson boundaries for fairly general classes of Markov chains on the Sierpiński graph. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.