On regular Dirichlet subspaces of \(H^1(I)\) and associated linear diffusions
type: Item
instance of: scholarly article
title: On regular Dirichlet subspaces of \(H^1(I)\) and associated linear diffusions
zbMATH Open document ID: 1068.60093
author: Zing Fang
author: Masatoshi Fukushima
author: Jiangang Ying
published in: Osaka Journal of Mathematics
publication date: 2005-05-06T00:00:00Z
review text: Let \(I\) denote either a bounded open interval or \(\mathbb R\). Starting with the classical (regular, recurrent, local) Dirichlet space \[ H^ {1} (I) := \{ u\in L ^ {2} (I) \mid u \text{ absolutely continuous},\;u' \in L ^{2} (I) \}, \] \[ \mathbf D (u, v) := \int _ {I} u' v' d x \] the authors describe completely the regular Dirichlet subspaces of this space. Also, a description of the recurrent regular Dirichlet subspaces, for the case \(I = \mathbb{R}\) is given. The subspaces are indexed by the set of functions \(s\) on \(I\), which are: absolutely continuous, strictly increasing, \(s'(x) = 0\) or \(1\) a.e. on \(I\), positive on \(I\), resp. on \((0, +\infty )\) if \( I = \mathbb{R}\). In the case \(I = \mathbb{R}\), the recurrent regular subspaces correspond with those \(s\) satisfying moreover \(s(\pm \infty ) = \pm \infty \). The associated process is constructed from the reflected Brownian motion, by a time change and a state space transformation.
reviewed by: Liliana Popa
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 31B15
zbMATH DE Number: 2165811
zbMATH Keywords: Dirichlet space
zbMATH Keywords: reflecting Brownian motion
zbMATH Keywords: local
zbMATH Keywords: regular
zbMATH Keywords: recurrent
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2165811