Induced measures of simple random walks on Sierpinski graphs

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}$}{{\bf x}}{\lim{n \rightarrow \infty} Z_n \in \cdot}$inducedon$K$\mathrm{.}Usingareflectionprinciplebasedonthesymmetriesof$K$, 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 $v_{{\bf x}}$,${\bf x} \in X$,isabsolutelycontinuouswithrespectto$\mu$. This answers a question of Kaimanovich [K, Problem 4.14]. The argument can be generalized to other symmetric self-similar sets.