Energy functionals and killing transform
type: Item
instance of: scholarly article
title: Energy functionals and killing transform
zbMATH Open document ID: 0892.60083
DOI: 10.1007/BF02882564
author: Jiangang Ying
published in: Chinese Science Bulletin
publication date: 1998-08-09T00:00:00Z
review text: In the standard setting of potential theory for right Markov process due to Getoor and Sharpe, the author gives a probabilistic interpretation of energy functional w.r.t. the subprocess killed by a given exact multiplicative functional. The main theorem is stated as follows: Let \({\mathbf X}= (X_t, P^x)\) be a right Markov process and denote respectively by \(\text{Exc}({\mathbf X})\), \(\text{S}({\mathbf X})\) the classes of excessive measures and excessive functions associated with \({\mathbf X}\). The energy-functional \(L:\text{Exc}({\mathbf X})\times\text{S}({\mathbf X})\to\mathbb{R}\) (of \(X\)) is by definition \(L(m,u):= \sup\{\mu(u): \mu\geq 0\), \(\mu{\mathbf U}\leq m\}\), where \({\mathbf U}\) denotes the potential-operator. By the same way if \(M\) is a given exact multiplicative functional with positive life time, then the energy-functional of the \(M\)-subprocess will be denoted by \(L^M(m,u)\). Theorem. 1) \(L(m,u) = L^M(m-{\mathbf R}_Mm, u)\), where \({\mathbf R}_M\) denotes the balayage-operator for measures. 2) Furthermore, if \(m\) is dissipative, then \(L(m,u)= L^M(m,u- {\mathbf P}_Mu)\), where \({\mathbf P}_M\) denotes the balayage-operator for functions. On the way to show the above theorem the author gives an interpretation of the Feynman-Kac formula in terms of energy-functional.
Mathematics Subject Classification ID: 60J45
zbMATH DE Number: 1106390
zbMATH Keywords: potential theory
zbMATH Keywords: right Markov process
zbMATH Keywords: energy-functional
zbMATH Keywords: multiplicative functional
zbMATH Keywords: balayage-operator
zbMATH Keywords: Feynman-Kac formula
reviewed by: Xuan Loc Nguyen
MaRDI profile type: Publication
cites work: Q4265489
cites work: Killing a Markov process under a stationary measure involves creation
cites work: Q4040434
cites work: Q3822953
cites work: Excessive measures and Markov processes with random birth and death
cites work: Revuz measures and time changes
full work available at URL: https://doi.org/10.1007/bf02882564
OpenAlex ID: W2044977919
description: scientific article; zbMATH DE number 1106390