Convergences and projection Markov property of Markov processes on ultrametric spaces

Let (S, ρ) be an ultrametric space satisfying certain conditions and Sk be the quotient space of S with respect to the partition by balls with a fixed radius φ(k). We prove that, for a Hunt process X on S associated with a Dirichlet form (ε, F), a Hunt process X^k on S^k associated with the averaged Dirichlet form (ε^k, 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 π^k to S^k. In this case, we see that the projected process π^k ◦ X is identical in law to X^k and converges almost surely to X.