Approximation of jump processes on fractals

Abstract

The article studies approximations for stable like jump processes on fractal sets $F\subset\mathbb{R}^{n}$. Processes on $d$-sets are approximated by jump processes on the $\varepsilon$-parallel sets. For the special case of self-similar sets with equal contraction ratios, approximations in terms of finite Markov chains are provided. In either case, the convergence of Dirichlet forms, semigroups and resolvents are established as well as the convergence of the finite-dimensional distributions under canonical initial distributions. In the self-similar case also the weak convergence of the laws under these initial distributions in $D_{F}([0,t_{0}])$ is proved.