Skip to main content

L2 convergence of certain random walks on Zd and related diffusions

Workshop Contributions

  • 376 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 1212)

Abstract

One technique for studying the approach to equilibrium of a continuous time Markov process is to consider the restriction to the L 2 space of an invariant distribution. When the process is reversible with respect to this distribution, the generator is a selfadjoint operator. We study the L 2 spectrum of the generator for certain random walks on Z d, where the reversible invariant distribution is concentrated near the origin and decays rapidly with distance to the origin. For the related diffusions on R d we find that the generators are unitarily equivalent to Schrödinger operators.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   44.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   59.95
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Buffet, E., Pulé, J., de Smedt, P.: On the dynamics of Bose-Einstein condensation. Ann. Inst. Henri Poincaré Analyse non-linéaire 1 413–451(1984)

    MathSciNet  MATH  Google Scholar 

  2. Hille, E. and Phillips, R.S.: Functional Analysis and Semi-groups, 2nd ed. Providence: AMS (1957)

    MATH  Google Scholar 

  3. Reed, M. and Simon, B.: Methods of Modern Mathematical Physics, Vol. 4. New York: Academic Press (1978)

    MATH  Google Scholar 

  4. Sullivan, W.G.: Mean square relaxation times for evolution of random fields. Commun. Math. Phys. 40 249–258 (1975)

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Sullivan, W.G.: Exponential convergence in dynamic Ising models with distinct phases. Phys. Let. 53A 441–2 (1975)

    CrossRef  Google Scholar 

  6. Sullivan,W.G.: The L 2 spectral gap of certain positive recurrent Markov chains and jump processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete (in press)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1986 Springer-Verlag

About this paper

Cite this paper

Sullivan, W.G. (1986). L2 convergence of certain random walks on Zd and related diffusions. In: Tautu, P. (eds) Stochastic Spatial Processes. Lecture Notes in Mathematics, vol 1212. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0076253

Download citation

  • DOI: https://doi.org/10.1007/BFb0076253

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16803-4

  • Online ISBN: 978-3-540-47053-3

  • eBook Packages: Springer Book Archive