Gaussian bounds, strong ellipticity and uniqueness criteria
type: Item
instance of: scholarly article
title: Gaussian bounds, strong ellipticity and uniqueness criteria
zbMATH Open document ID: 1311.47031
author: Derek W. Robinson
published in: Bulletin of the London Mathematical Society
publication date: 2014-10-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/1401.0360
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 47D07
zbMATH DE Number: 6355605
zbMATH Keywords: second order elliptic operator
zbMATH Keywords: parabolic operator
zbMATH Keywords: selfadjoint operator
zbMATH Keywords: Markov semigroup
zbMATH Keywords: diffusion process
zbMATH Keywords: Gaussian kernel
MaRDI profile type: Publication
OpenAlex ID: W2094280606
arXiv ID: 1401.0360
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DOI: 10.1112/BLMS/BDU063
review text: Consider a quadratic form \(h(\varphi)=\sum_{i,j=1}^d (\partial_i\varphi, c_{ij}\partial_j\varphi)\) defined on \(W^{1,2}(\mathbb{R}^d),\) where \(c_{ij}=c_{ji}\) are real-valued, locally bounded and measurable functions such that the matrix \(C=\{c_{ij}\}\) is non-negatively defined. If \(C\) is strongly elliptic, that is, \(0<\mu I\leq C\leq \lambda I\) for \(\lambda,\mu>0,\) then \(h\) is closable, and the closure determines a positive self-adjoint operator \(H\) on \(L^2(\mathbb{R}^d)\) which generates a submarkovian semigroup \(S\) with a positive distributional kernel \(K\) and this kernel satisfies Gaussian upper and lower bounds. Moreover, \(S\) is conservative, that is, \(S_t1=1\) for all \(t>0\).NEWLINENEWLINEIn the paper under review, the author examines the converse statements. First, it is established that \(C\) is strongly elliptic if and only if \(h\) is closable, the semigroup \(S\) is conservative and \(K\) satisfies Gaussian bounds. Secondly, the author proves that, if the coefficients \(c_{ij}\) are such that a Tikhonov growth condition is satisfied, then \(S\) is conservative. Thus, in this case strong ellipticity of \(C\) is equivalent to closability of \(h\) together with Gaussian bounds for \(K\). Finally, taking the coefficients in \(W^{1,\infty}_{\text{loc}}(\mathbb{R}^d),\) it is shown that \(h\) is closable and a growth condition of the Täcklind type is sufficient to establish the equivalence of strong ellipticity of \(C\) and Gaussian bounds on \(K.\)
reviewed by: Dian K. Palagachev
description: scientific article; zbMATH DE number 6355605