Convergences and Projection Markov Property of Markov Processes on Ultrametric Spaces

Abstract

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 $\varphi (k)$. 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 almost surely to $X$.