Jump processes on leaves of multibranching trees

TL;DR

This paper explores stochastic processes on hierarchical trees by extending p-adic number applications, focusing on a new sequence space with a non-Archimedean metric suitable for variable-branching trees.

Contribution

It introduces a novel sequence space with a non-Archimedean metric for labeling variable-branching trees and extends stochastic process constructions from p-adics to this space.

Findings

The sequence space has well-defined topological properties.

Stochastic processes on p-adics can be generalized to the new space.

The approach applies to trees with varying numbers of branches.

Abstract

The p-adic numbers have found applications in a wide range of diverse fields of research. In some applications the algebraic properties of p-adics enter as an indispensable ingredient of the theory. Another class of applications has to do with hierarchical tree like systems. In this context the applications are based on the well known correspondence between p-adics and the trees with p-branches emerging from every branching point. Then the algebraic structure does not enter and p-adics are used merely as a labeling system for the tree branches. We introduce a space of numerical sequences suitable for labeling the trees with varying number of branches emerging from the branching points. We equipe this space with a non Archimedian metric and describe its basic topological properties. We also demonstrate that the known construction of the stochastic processes on p-adics carry over to the stochastic processes on the above mentioned space and hence on the corresponding trees.