Conditional gauge theorem for non-local Feynman-Kac transforms
type: Item
instance of: scholarly article
title: Conditional gauge theorem for non-local Feynman-Kac transforms
zbMATH Open document ID: 1025.60032
author: Zhen-Qing Chen
author: Renming Song
publication date: 2003-05-04T00:00:00Z
review text: Let \(X=(X_t,P_x)\) be a transient Borel standard process on a Lusin space \(E\). Further assume that, for an excessive measure \(m\) on \(E\), \(X\) is \(m\)-irreducible having Green function \(G(x,y)\). The main purpose of this paper is to prove the gauge and the conditional gauge theorem related to discontinuous additive functionals. For two Borel functions \(q\) on \(E\) and \(F\) on \(E\times E\) vanishing on the diagonal, let \[ A_{q+F} (t)=\int^t_0 q(X_s)ds+ \sum_{s\leq t}F(X_{s-},X_s) \quad\text{and}\quad e_{q+F} (t)=\exp (A_{q+F} (t)). \] The function \(q\) is said to be of class \({\mathbf K}_1\) if \(\beta(q)= \sup_{B \subset K,m(B)< \delta}\sup_x G(|q|I_{K^c\cup B}) <1\) for some \(K\) and \(\delta >0\). Also \(F\) is said a function of class \({\mathcal I}_\infty\) if \(F\) is bounded and \(N|F|(x) \mu_H(dx)\) is a Kato type measure, where \((N,H)\) is the Lévy system and \(\mu_H\) is the Revuz measure of \(H\). The gauge theorem in this paper says that, for \(q\in{\mathcal K}_1\) and \(F\in{\mathcal I}_\infty\), \(E_x (e_{g+F}(\zeta))\) is either bounded or identically infinite. When the dual process exists, the corresponding classes \({\mathbf S}_1\) and \({\mathbf A}_\infty\) related to the \(h\)-process \(X^z= (X_t,P^z_x)\) for \(h=G( \cdot,z)\) are introduced. For \(q\in {\mathbf S}_1\) and \(F\in {\mathbf A}_\infty\), it is proved that \(E^z_x(e_{q+F} (\zeta^z))\) is bounded if it is finite at some \((x_0,z_0)\). Furthermore, an expression of the bilinear form of the process determined by a Feynman-Kac transformation corresponding to \(e_{q+F}(t)\) is given.
reviewed by: Yoichi Oshima
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 60J40
Mathematics Subject Classification ID: 35J10
zbMATH DE Number: 1903768
zbMATH Keywords: gauge theorem
zbMATH Keywords: Lévy system
zbMATH Keywords: Feynman-Kac transform
MaRDI profile type: Publication
full work available at URL: s004400200219
OpenAlex ID: W2021278879
DOI: 10.1007/S004400200219
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1903768