Approximation of jump processes on fractals
The paper is devoted to the approximations for jump processes on \(d\)-sets, in particular, on self-similar sets. Starting with the set of Dirichlet forms on approximating \(\varepsilon\)-sets, it is shown that there exists the Mosco--limit Dirichlet form (in the sense of \textit{K. Kuwae} and \textit{T. Shioya} [Commun. Anal. Geom. 11, No.~4, 599--673 (2003; Zbl 1092.53026)]). Then the strong convergence (in the sense of Kuwae-Shioya) of the resolvents and the operator semigroups naturally follows by the techniques from [Kuwae and Shioya, loc. cit.] and [\textit{U. Mosco}, J. Funct. Anal. 123, No.~2, 368--421 (1994; Zbl 0808.46042)]. In the case of a self-similar set, it is proved that the approximate and limit forms satisfy the Nash's inequality, which implies the existence of transition densities for both the approximating and the limit processes. To prove the tightness the estimate on the first exit time from a ball of certain radius is proved, which together with \textit{D. Aldous}' Theorem [Ann. Probab. 6, 335--340 (1978; Zbl 0391.60007)] implies the weak convergence of the related processes in Skorokhod space.
scientific article; zbMATH DE number 5543714
scientific article