Markov chain approximations to symmetric diffusions (Q1372355)

A Markov chain approximation for diffusions associated to divergence form operators is considered. The present paper aims to the case of generators with nonsmooth coefficients, in which the identification of limits via Dirichlet forms, which is a basic tool for symmetric Markov processes from divergence form operators, is rather hard. A number of delicate estimations is given first, which leads to a discrete setting of the De Giorgi-Moser-Nash theory. Then, a concrete symmetric Markov chain approximation for symmetric diffusions is found. This approximation for diffusions is different from that in \textit{D. W. Stroock} and \textit{S. R. S. Varadhan}'s book ``Multidimensional diffusion theory'' (1979; Zbl 0426.60069), where the approximate Markov chains are time discrete, while in the present paper the approximate Markov chains are discrete in space.