H.KanekoAn Orlicz space on ends of tree and superposition of nodewise given Dirichlet forms with tierlinkageAbstract
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.
