$p$-ADICS.2015, 07-12.09.2015, Belgrade, Serbia

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An Orlicz space on ends of tree and superposition of nodewise given Dirichlet forms with tier-linkage


Stochastic processes on the field of $p$-adic numbers have been studied for more than two decades. The theory of Dirichlet space is applied in important parts of the development in the subject. In some recent observations, hierarchical structure is determined by tree structure. It turns out that and the design of tree determines spectral characteristics of measure symmetric stochastic processes on the ends of a tree so that it provides us with an enlarged framework of the stochastic analysis on $p$-adics. Those researches were made by focusing on the family of eigenfunctions which coincides with the complete orthogonal system in the space of square integrable functions on the ends of the tree. Such coincidence is mainly due to absence of linkage between distinct nodes in some stochastic sense. The objective of the present talk is dealing with the case where a Dirichlet space is assigned at every node and linkages between them are described by other family of bilinear forms determined by pairs of nodes connected by a branch or shared by an identical parental node. This means that we will need to consider the case where the structure of eigenfunctions differs from the one in existing studies. In fact, we are going to deal with the complete orthogonal system in the space of square integrable functions which does not coincide with the family of eigenfunctions. In the final part, we will discuss the lower estimate on capacity on the space of the ends based on Orlicz space theory in the case such linkages are allowed. These stories can be traced back to J.L.Doob who investigated a jump process on the unit circle traveling points taken by reflections of the reflected Brownian motion on the unit disk.