Potential theory related to some multiparameter processes
type: Item
instance of: scholarly article
title: Potential theory related to some multiparameter processes
zbMATH Open document ID: 0839.31010
DOI: 10.1007/BF01071696
author: Francis Hirsch
published in: Potential Analysis
publication date: 1996-06-19T00:00:00Z
review text: In the first part a potential theory is constructed from a continuous kernel on a locally compact metrizable space. This includes a study of capacity, quasi-continuity, equilibrium measures and potentials. In the second part the author presents the example of a kernel \(U\) on \(\mathbb{R}^d\), \(d\geq 1\), which is defined using \(n\) convolution semigroups \((\pi^j_t)_{t \geq 0}\), \(1 \leq j \leq n\), of symmetric probability measures on \(\mathbb{R}^d\) \[ Uf = f* v_1* \cdots* v_n \quad \text{where} \quad v_j = (1/ \sqrt \pi) \int^\infty_0 t^{-1/2} e^{-t} \pi^j_t dt. \] In the third and main part the probabilistic counterpart is discussed in detail. It is a multiparameter process \((Z_t^x)_{t \in \mathbb{R}^n_+}\) where \(Z^x_{(t_1, \dots, t_n)} = x + X^1_{t_1} + \cdots + X^n_{t_n}\) and \(X^1, \dots, X^n\) are \(n\) independent processes given by \((\pi^1_t)_{t \geq 0}, \dots, (\pi^n_t)_{t \geq 0}\), respectively.
reviewed by: Wolfhard Hansen
Mathematics Subject Classification ID: 31C15
Mathematics Subject Classification ID: 60J99
Mathematics Subject Classification ID: 60J45
zbMATH DE Number: 763780
zbMATH Keywords: capacity
zbMATH Keywords: quasi-continuity
zbMATH Keywords: equilibrium measures
zbMATH Keywords: multiparameter process
MaRDI profile type: Publication
cites work: Additive functionals of several time-reversible Markov processes
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cites work: Occupation densities
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cites work: Quasi-continuous functions and Hunt processes
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description: scientific article; zbMATH DE number 763780