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
The asymptotic behavior of the principal eigenvalue in a singular perturbation problem with invariant boundaries
Download PDF
Download PDF
  • Published: December 1987

The asymptotic behavior of the principal eigenvalue in a singular perturbation problem with invariant boundaries

  • Alexander Eizenberg1 &
  • Yuri Kifer2 

Probability Theory and Related Fields volume 76, pages 439–476 (1987)Cite this article

  • 115 Accesses

  • 8 Citations

  • Metrics details

Summary

We consider diffusion random perturbations of a dynamical systemS t in a domainG⊂R m which, in particular, may be invariant under the action ofS t. Continuing the study of [K1-K4] we find the asymptotic behavior of the principal eigenvalue of the corresponding generator when the diffusion term tends to zero.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  • Aronson, D.G.: The fundamental solution of a linear parabolic equation containing a small parameter. Illinois J. Math.3, 580–619 (1959)

    Google Scholar 

  • Cornfeld, I.P., Fomin, S.V., Sinai, Ya.G.. Ergodic theory. Berlin, Heidelberg, New York: Springer 1982

    Google Scholar 

  • Devinatz, A., Ellis, R., Friedman, A.: The asymptotic behavior of the first real eigenvalue of second order elliptic operators with a small parameter in the highest derivatives. Indiana Univ. Math. J.23, 991–1011 (1974)

    Google Scholar 

  • Eizenberg, A.: The exit distribution for small random perturbations of dynamical systems with a repulsive type stationary point. Stochastics12, 251–275 (1984)

    Google Scholar 

  • Freidlin, M.. Functional integration and partial differential equations. Princeton: Princeton Univ. Press 1985

    Google Scholar 

  • Friedman, A.: Stochastic differential equations and applications. New York: Academic Press 1975

    Google Scholar 

  • Friedman, A: The asymptotic behavior of the first real eigenvalue of a second order elliptic operator with small parameter in the highest derivatives. Indiana Univ. Math. J.22, 1005–1015 (1973)

    Google Scholar 

  • Hartman, P.: Ordinary differential equations. New York: Wiley 1964

    Google Scholar 

  • Hirsh, M.W., Pugh, C.C., Shub, M.: Invariant manifolds, Lecture Notes Math. 583. Berlin, Heidelberg, New York: Springer 1977

    Google Scholar 

  • Kifer, Yu: On the principal eigenvalue in a singular perturbation problem with hyperbolic limit points and circles. J. Differ. Equations37, 108–139 (1980)

    Google Scholar 

  • Kifer, Yu.: Stochastic stability of the topological pressure. J. D'Analyse Math.38, 255–286 (1980)

    Google Scholar 

  • Kifer, Yu.: The exit problem for small random perturbations of dynamical systems with a hyperbolic fixed point. Isrtael J. Math.40, 74–96 (1981)

    Google Scholar 

  • Kifer, Yu.: The inverse Problem for small random perturbations of dynamical systems. Israel J. Math.40, 165–174 (1981)

    Google Scholar 

  • Lind, D.A.: Spectral invariants in smooth ergodic theory. Lecture Notes in Phys.38, 296–308 (1975)

    Google Scholar 

  • Varadhan, S.R.S.: Lectures on diffusion problems and partial differential equations, Tata Inst. of Fund. research, Bombay. Berlin, Heidelberg, New York: Springer 1980

    Google Scholar 

  • Ventcel, A.D.: On the asymptotic behavior of the greatest eigenvalue of a second-order elliptic differential operator with a small parameter in the higher derivatives. Soviet Math. Dokl.13, 13–17 (1972)

    Google Scholar 

  • Ventcel, A.D., Freidlin, M.I.: On small random perturbations of dynamical systems. Russ. Math. Surv.25, 1–55 (1970)

    Google Scholar 

  • Walters, P.: An introduction to ergodic theory. New York: Springer 1982

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Tel-Aviv University, Ramat Aviv, Israel

    Alexander Eizenberg

  2. Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel

    Yuri Kifer

Authors
  1. Alexander Eizenberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuri Kifer
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This work was supported by U.S.A.-Israel B.S.F. Grant #84-00028

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Eizenberg, A., Kifer, Y. The asymptotic behavior of the principal eigenvalue in a singular perturbation problem with invariant boundaries. Probab. Th. Rel. Fields 76, 439–476 (1987). https://doi.org/10.1007/BF00960068

Download citation

  • Received: 24 April 1986

  • Issue Date: December 1987

  • DOI: https://doi.org/10.1007/BF00960068

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

  • Stochastic Process
  • Asymptotic Behavior
  • Probability Theory
  • Mathematical Biology
  • Singular Perturbation
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