  Let $K$ be a self-similar set satisfying the open set condition. Following
Kaimanovich's elegant idea, it has been proved that on the symbolic space $X$
of $K$ a natural augmented tree structure ${\mathfrak E}$ exists; it is
hyperbolic, and the hyperbolic boundary $\partial_HX$ with the Gromov metric is
H\"older equivalent to $K$. In this paper we consider certain reversible random
walks with return ratio $0< \lambda <1$ on $(X, {\mathfrak E})$. We show that
the Martin boundary ${\mathcal M}$ can be identified with $\partial_H X$ and
$K$. With this setup and a device of Silverstein, we obtain precise estimates
of the Martin kernel and the Na\"{i}m kernel in terms of the Gromov product.
Moreover, the Na\"{i}m kernel turns out to be a jump kernel satisfying the
estimate $\Theta (\xi, \eta) \asymp |\xi-\eta|^{-(\alpha+ \beta)}$, where
$\alpha$ is the Hausdorff dimension of $K$ and $\beta$ depends on $\lambda$.
For suitable $\beta$, the kernel defines a regular non-local Dirichlet form on
$K$. This extends the results of Kigami concerning random walks on certain
trees with Cantor-type sets as boundaries.
