Convergence of symmetric Markov chains on $\Z^d$
  For each $n$ let $Y^n_t$ be a continuous time symmetric Markov chain with state space $n^{-1} \Z^d$. A condition in terms of the conductances is given for the convergence of the $Y^n_t$ to a symmetric Markov process $Y_t$ on $\R^d$. We have weak convergence of $\{Y^n_t: t\leq t_0\}$ for every $t_0$ and every starting point. The limit process $Y$ has a continuous part and may also have jumps. 
R.F. Bass, T. Kumagai, and T. Uemura