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Induced measures of simple random walks on Sierpinski graphs
- Publication Year :
- 2010
-
Abstract
- In \cite{[K]}, Kaimanovich defined an augmented rooted tree $(X, E)$ corresponding to the Sierpinski gasket $K$, and showed that the Martin boundary of the simple random walk $\{Z_n\}$ on it is homeomorphic to $K$. It is of interest to determine the hitting distributions $v_{{\bf x}}(\cdot) = {\mathbb{P}}_{{\bf x}}\{\lim_{n \rightarrow \infty} Z_n \in \cdot\}$ induced on $K$. Using a reflection principle based on the symmetries of $K$, we show that if the walk starts at the root of $(X, E)$, the hitting distribution is exactly the normalized Hausdorff measure $\mu$ on $K$. In particular, each $v_{{\bf x}}$, ${\bf x} \in X$, is absolutely continuous with respect to $\mu$. This answers a question of Kaimanovich [K, Problem 4.14]. The argument can be generalized to other symmetric self-similar sets.<br />Comment: 19 pages, 10 figures
- Subjects :
- Mathematics - Probability
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1012.5347
- Document Type :
- Working Paper