Equilibrium measures on recurrent Markov processes
type: Item
instance of: scholarly article
title: Equilibrium measures on recurrent Markov processes
zbMATH Open document ID: 0583.60072
author: Kumiko Kitamura
published in: Proceedings of the Japan Academy. Series A
publication date: 1985-01-01T00:00:00Z
review text: The equilibrium measures in Newtonian potential theory were interpreted by \textit{K. L. Chung} [Ann. Inst. Fourier 25(1975), No.3-4, 131-138 (1976; Zbl 0338.31001)] and others in terms of the hitting-distributions of the associated standard Brownian motion. In the present note the author treats the same problem for the case of recurrent Markov processes. The associated Green function on the given open set \(L^ c\) is supposed to be symmetric. The main results are the following: a) The equilibrium measure \(\nu^ K_ L\) of a set \(K\subset L^ c\) can be identified as \(\nu^ K_ L=\mu^ K_ L\cdot C(K,L)\) where \(\mu^ K_ L\) and C(K,L) are the concepts measure and capacity introduced earlier by \textit{T. Ueno} [Kodai Math. Sem. Rep. 12, 109-142 (1960; Zbl 0094.322)] for recurrent processes. b) Furthermore if the Green function \(g_ L(\cdot,\cdot)\) is lower semi- continuous then \(g_ L\nu^ K_ L\) could be interpreted as the distribution of the hitting-time of the process for the set \(K\) before leaving \(L^ c.\) Details of the statements are too long to be given here.
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 60J25
zbMATH DE Number: 3934157
zbMATH Keywords: equilibrium measures
zbMATH Keywords: Green function
zbMATH Keywords: recurrent processes
reviewed by: Xuan Loc Nguyen
MaRDI profile type: Publication
full work available at URL: pjaa.61.43
OpenAlex ID: W1995562201
cites work: Q5556844
cites work: Q3946912
cites work: Q3048019
cites work: On recurrent Markov processes
DOI: 10.3792/PJAA.61.43
description: scientific article; zbMATH DE number 3934157