Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise
Download PDF
Download PDF
  • Published: April 2000

Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise

  • J. Martin Lindsay1 &
  • Stephen J. Wills1 

Probability Theory and Related Fields volume 116, pages 505–543 (2000)Cite this article

  • 112 Accesses

  • 32 Citations

  • Metrics details

Abstract.

Quantum stochastic differential equations of the form

govern stochastic flows on a C *-algebra ?. We analyse this class of equation in which the matrix of fundamental quantum stochastic integrators Λ is infinite dimensional, and the coefficient matrix θ consists of bounded linear operators on ?. Weak and strong forms of solution are distinguished, and a range of regularity conditions on the mapping matrix θ are considered, for investigating existence and uniqueness of solutions. Necessary and sufficient conditions on θ are determined, for any sufficiently regular weak solution k to be completely positive. The further conditions on θ for k to also be a contraction process are found; and when ? is a von Neumann algebra and the components of θ are normal, these in turn imply sufficient regularity for the equation to have a strong solution. Weakly multiplicative and *-homomorphic solutions and their generators are also investigated. We then consider the right and left Hudson-Parthasarathy equations:

in which F is a matrix of bounded Hilbert space operators. Their solutions are interchanged by a time reversal operation on processes. The analysis of quantum stochastic flows is applied to obtain characterisations of the generators F of contraction, isometry and coisometry processes. In particular weak solutions that are contraction processes are shown to have bounded generators, and to be necessarily strong solutions.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK. e-mail: jml@maths.nott.ac.uk; sjw@maths.nott.ac.uk, , , , , , GB

    J. Martin Lindsay & Stephen J. Wills

Authors
  1. J. Martin Lindsay
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stephen J. Wills
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Received: 3 November 1998 / Published online: 30 March 2000

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Lindsay, J., Wills, S. Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise. Probab Theory Relat Fields 116, 505–543 (2000). https://doi.org/10.1007/s004400050261

Download citation

  • Issue Date: April 2000

  • DOI: https://doi.org/10.1007/s004400050261

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Weak Solution
  • Stochastic Differential Equation
  • Strong Solution
  • Bounded Linear Operator
  • Stochastic Integrator
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature