Approximation of jump processes on fractals
type: Item
instance of: scholarly article
title: Approximation of jump processes on fractals
zbMATH Open document ID: 1171.60019
author: Michael Hinz
published in: Osaka Journal of Mathematics
publication date: 2009-04-14T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1235574042
review text: 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.
Mathematics Subject Classification ID: 60J75
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 60J35
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 5543714
zbMATH Keywords: Mosco convergence
zbMATH Keywords: fractal sets
zbMATH Keywords: self-similar sets
zbMATH Keywords: Dirichlet forms
zbMATH Keywords: jump processes on fractal sets
reviewed by: V. Knopova
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5543714