A Dirichlet Space on Ends of Tree and Dirichlet Forms with a Nodewise Orthogonal Property

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 studies. In some recent observations, the hierarchical structure is handled by tree structure and the structure gives an overview for measure symmetric stochastic processes on the field of p-adic numbers and ends of a tree. An orthogonal property described in terms of Dirichlet form for eigenfunctions associated with distinct nodes is involved coherently in the existing formalisms based on Dirichlet space theory. The objective of the present article is paying a close attention to some characteristic property implied by the orthogonal property and presenting a wider formalism in accordance with the characteristic property.