Local and Non-Local Dirichlet Forms on the Sierpi\'nski Gasket and the Sierpi\'nski Carpet

Abstract

This thesis is about local and non-local Dirichlet forms on the Sierpi\'nski gasket and the Sierpi\'nski carpet. We are concerned with the following three problems in analysis on the Sierpi\'nski gasket and the Sierpi\'nski carpet. First, a unified purely \emph{analytic} construction of local regular Dirichlet forms on the Sierpi\'n-ski gasket and the Sierpi\'nski carpet. We give a purely analytic construction of a self-similar local regular Dirichlet form on the Sierpi\'nski carpet using $\Gamma$-convergence of stable-like non-local closed forms which gives an answer to an open problem in analysis on fractals. We also apply this construction on the Sierpi\'nski gasket. Second, determination of walk dimension \emph{without} using diffusion. Although the walk dimension is a parameter that determines the behaviour of diffusion, we give two approaches to the determination of the walk dimension \emph{prior} to the construction of diffusion. Third, approximation of local Dirichlet forms by non-local Dirichlet forms. We prove that non-local Dirichlet forms can approximate local Dirichlet forms as direct consequences of our construction of local Dirichlet forms. We also prove that on the Sierpi\'nski gasket the local Dirichlet form can be obtained as a Mosco limit of non-local Dirichlet forms. Let us emphasize that we do \emph{not} need subordination technique based on heat kernel estimates.