Induced measures of simple random walks on Sierpinski graphs

TL;DR

This paper proves that the hitting distributions of simple random walks on Sierpinski graphs are exactly the normalized Hausdorff measure, using symmetry and reflection principles, answering a question posed by Kaimanovich.

Contribution

It establishes the precise form of hitting distributions for random walks on Sierpinski graphs, showing they are absolutely continuous with respect to the Hausdorff measure, and extends the method to other self-similar sets.

Findings

Hitting distributions are the normalized Hausdorff measure on the Sierpinski gasket.

Hitting distributions are absolutely continuous with respect to the Hausdorff measure.

Method can be generalized to other symmetric self-similar fractals.

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.