Uniqueness and non-uniqueness of semigroups generated by singular diffusion operators
type: Item
instance of: scholarly article
title: Uniqueness and non-uniqueness of semigroups generated by singular diffusion operators
zbMATH Open document ID: 0957.60002
author: Andreas Eberle
published in: Lecture Notes in Mathematics
publication date: 2000-01-19T00:00:00Z
review text: Let \(E\) be an open subset of \(\mathbb{R}^d\). Consider a differential opertor on \(E\) of type \[ Lf= \sum_{i,j=1}^d a_{ij} \frac{\partial^2f} {\partial x_i\partial x_j}+ \sum_{j=1}^d b_j \frac{\partial f}{\partial x_j}, \quad f\in{\mathcal A}, \tag{1} \] where \(a_{ij},b_j:E\to \mathbb{R}\) are measurable coefficient functions so that \((a_{ij})_{i,j}\) is symmetric and nonnegative definite, and \({\mathcal A}\) is a space of smooth functions (e.g. \({\mathcal A}= C_0^\infty(E)\)). Suppose that there exists a positive measure \(m\) on \(E\) and \(\alpha> 0\), such that \[ \int Lf dm\leq \alpha\int f dm \quad \text{for all }f\in{\mathcal A},\;f\geq 0,\tag{2} \] where the integrals are assumed to exist. The monograph analyzes the question whether the operator \((L,{\mathcal A})\) has a unique extension \((\widehat{L}, D(\widehat{L}))\) which generates a \(C_0\)-semigroup \(T_t= e^{t\widehat{L}}\), \(t> 0\), on \(L^p (E,m)\) (provided, of course, \(p\in [1,\infty)\) is such that \((L,{\mathcal A})\) is an operator on \(L^p (E,m)\)). One should note that analyzing the operator on \(L^p(E,m)\) rather than the classical \(L^p\)-space with respect to Lebesgue measure is in many cases much more convenient due to the fact that measures as in (2) (like e.g. infinitesimally invariant measures) are much more adapted to \(L\) than Lebesgue measure. There is a vast number of existence results for measures satisfying (2) for a given \(L\). The motivation for examining such kind of uniqueness questions for example comes from the interest in solving the Cauchy problem for \(L\) (whose solution is given by \(e^{t\widehat{L}}\), \(t> 0\)) or from probability theory, since in many cases \(e^{t\widehat{L}}\), \(t>0\), describes the transition probabilities of a Markov process. Non-uniqueness for such extensions \(\widehat{L}\) is, of course, well-known to come from different possible boundary conditions. But there are also two more causes for non uniqueness. One is the singularity o…
Mathematics Subject Classification ID: 60-02
Mathematics Subject Classification ID: 31C25
Mathematics Subject Classification ID: 31-02
Mathematics Subject Classification ID: 60J60
zbMATH DE Number: 1391000
zbMATH Keywords: singular diffusion operators
zbMATH Keywords: \(C_0\)-semigroups
zbMATH Keywords: essential selfadjointness
zbMATH Keywords: Dirichlet forms
zbMATH Keywords: Markov uniqueness
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0103045
OpenAlex ID: W146490786
DOI: 10.1007/BFB0103045
description: scientific article; zbMATH DE number 1391000