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Brownian motion penetrating fractals. An application of the trace theorem of Besov spaces. (English) Zbl 0960.31005

Let \(F\) be a closed connected set in \(\mathbb{R}^n\) and \(\mu\) a positive Borel measure on \(F\). Let \(({\mathcal E}, D({\mathcal E}))\) be a local regular Dirichlet form on \(L^2(F, \mu)\) so that \(D({\mathcal E})\) is contained in a Lipschitz or Besov space.
The first main result of this really interesting paper is that under some assumptions on \(\mu\) there is a local regular Dirichlet form \((\widetilde{{\mathcal E}}, D(\widetilde{{\mathcal E}}))\) on \(L^2(\mathbb{R}^n, \mu+ \lambda)\), where \(\lambda\) denotes Lebesgue measure, whose restriction to \(F\) is \(({\mathcal E}, D({\mathcal E}))\) and which is the classical Dirichlet form on \(\mathbb{R}^n\setminus F\) associated to the Laplacian. So, by Fukushima’s theory of regular Dirichlet forms there exists a corresponding diffusion process on \(\mathbb{R}^n\).
The second main result of the paper under consideration states that under some assumptions on the order of the Lipschitz resp. Besov space in relation to properties of \(\mu\), the Dirichlet form \((\widetilde{{\mathcal E}}, D(\widetilde{{\mathcal E}}))\) is irreducible and the diffusion penetrates \(F\), provided the Newtonian 1-capacity of \(F\) is strictly positive. Applications include the cases where \(F\) is a nested fractal or the Sierpinski carpet.

MSC:

31C25 Dirichlet forms
60J60 Diffusion processes
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
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