Multiplicative matrix-valued functionals and the continuity properties of semigroups corresponding to partial differential operators with matrix-valued coefficients
type: Item
instance of: scholarly article
title: Multiplicative matrix-valued functionals and the continuity properties of semigroups corresponding to partial differential operators with matrix-valued coefficients
zbMATH Open document ID: 1223.47044
author: B. Güneysu
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-05-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/1105.0748
review text: The paper is concerned with probabilistic methods for selfadjoint operators in \(L^2({\mathbb R}^n, {\mathbb C}^d)\) that are formally given by the differential expression \[ \tau(\alpha, V)= -\frac{1}{2}\Delta - \frac{1}{2}\sum_{j=1}^n \alpha^2_j -\frac{1}{2}\sum_{j=1}^n (\partial_j\alpha_j ) -\sum_{j=1}^n \alpha_j\partial_j +V, \] where \(\alpha\) is a 1-form on \(\mathbb R^n\) and \(V\) is a potential. The author proves that the semigroup associated to this partial differential operator maps \(L^2({\mathbb R}^n, {\mathbb C}^d)\) to the space of continuous bounded functions and has a jointly continuous and spatially bounded integral kernel. Differential operators of this type arise in nonrelativistic mechanics when one wants to describe the energy of Yang-Mills particles which live on \(\mathbb R^n\) under the influence of the electrical potential \(V\).
reviewed by: Ruhollah Jahanipur
Mathematics Subject Classification ID: 47D08
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 47F05
zbMATH DE Number: 5888557
zbMATH Keywords: Feynman-Kac formula
zbMATH Keywords: Schrödinger operators
zbMATH Keywords: stochastic differential equations
zbMATH Keywords: partial differential equations
MaRDI profile type: Publication
OpenAlex ID: W2964229639
arXiv ID: 1105.0748
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DOI: 10.1016/J.JMAA.2011.02.038
description: scientific article