Convergences and projection Markov property of Markov processes on ultrametric spaces

Let $(S,\rho)$ be an ultrametric space with certain conditions and $S^k$ be the quotient space of $S$ with respect to the partition by balls with a fixed radius $\phi(k)$. We prove that, for a Hunt process $X$ on $S$ associated with a Dirichlet form $(\mathcal E, \mathcal F)$, a Hunt process $X^k$ on $S^k$ associated with the averaged Dirichlet form $(\mathcal E^k, \mathcal F^k)$ is Mosco convergent to $X$, and under certain additional conditions, $X^k$ converges weakly to $X$. Moreover, we give a sufficient condition for the Markov property of $X$ to be preserved under the canonical projection $\pi^k$ to $S^k$. In this case, we see that the projected process $\pi^k\circ X$ is identical in law to $X^k$ and converges weakly to $X$.