Classical stochastic processes from quantum stochastic calculus
type: Item
instance of: scholarly article
title: Classical stochastic processes from quantum stochastic calculus
zbMATH Open document ID: 0992.81041
DOI: 10.1023/A:1011322616457
published in: Journal of Mathematical Sciences (New York)
publication date: 2002-06-16T00:00:00Z
review text: Using multidimensional quantum stochastic calculus, the author constructs a (weak) one-parameter representation \(\tilde{j}_t\), \(t\in {\mathbb R}_+\), of the Lie algebra \(gl(N)\) of \(N\times N\) matrices, which is then extended to a representation \(\tilde{J}_t\), \(t\in {\mathbb R}_+\), of the universal enveloping algebra \({\mathcal U}(gl(N))\) of \(gl(N)\). Given a self-adjoint Casimir element \(Z\) in the center \({\mathcal Z} ({\mathcal U}(gl(N)))\) of \({\mathcal U}(gl(N))\), a classical stochastic process in the vacuum state is constructed as \(Z_t = \tilde{J}_tZ\), \(t\in {\mathbb R}_+\). When \(Z\) is a symmetric polynomial one obtains new integer-valued classical processes, and the classical Poisson process as a particular case. Chaotic expansions in multiple quantum stochastic integrals are also obtained for these processes.
reviewed by: Nicolas Privault
Mathematics Subject Classification ID: 81S25
Mathematics Subject Classification ID: 46N50
Mathematics Subject Classification ID: 60H05
zbMATH DE Number: 1756305
zbMATH Keywords: quantum stochastic calculus
zbMATH Keywords: universal enveloping algebras
zbMATH Keywords: Casimir elements
zbMATH Keywords: classical processes
zbMATH Keywords: Poisson process
author: R. L. Hudson
MaRDI profile type: Publication
full work available at URL: a:1011322616457
OpenAlex ID: W105604428
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description: scientific article; zbMATH DE number 1756305