Brownian motion penetrating fractals. An application of the trace theorem of Besov spaces
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.
scientific article; zbMATH DE number 1416440