Vadim A. Kaimanovich
Random Walks on Sierpi\'nski Graphs: Hyperbolicity and Stochastic Homogenization
Preprint series: ESI preprints

MSC:

60J10 Markov chains with discrete parameter

05C75 Structural characterization of types of graphs

28A80 Fractals, See also {58Fxx}

31C35 Martin boundary theory, See also {60J50}

28D05 Measure-preserving transformations

53C23 Global topological methods (a la Gromov)

60J50 Boundary theory

Abstract: 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 \Sie gasket (the associated
hyperbolic graph is called the \Sie graph), we show that the \Sie gasket can be
identified with the Martin and the Poisson boundaries for fairly general
classes of Markov chains on the \Sie graph.
Keywords: Sierpinski gasket, random walk, hyperbolic graph, Martin boundary, harmonic function, equivalence relation, harmonic measure