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
Random walks with killing
Download PDF
Download PDF
  • Published: February 1989

Random walks with killing

  • Neal Madras1 nAff2 

Probability Theory and Related Fields volume 80, pages 581–600 (1989)Cite this article

Summary

Let P(x, dy) be the transition probability kernel of a random walk on R n.Let k: R n→[0,1] be a given function. Suppose that the random walk is “killed” according to k: if it is at position x, then it jumps to a special state (the cemetery) with probability k(x); otherwise, it executes a usual random walk step (with law given by P). We say that the killed process is “long-lived” if:

  1. (a)

    P is transient, and there is a non-zero probability of never jumping to the cemetery; or if

  2. (b)

    P is recurrent, and the process started “from infinity” has a chance to get close to the origin before being killed.

This paper develops potential-theoretic characterizations of those functions k which permit long-lived behavior.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Bronner, F.: Frontiere de Martin d'un processus recurrent au sens de Harris. Z. Wahrscheinlich-keitstheor. Verw. Geb. 44, 227–251 (1978)

    Google Scholar 

  2. Brunel, A., Revuz, D.: Marches Recurrentes au sens de Harris sur les groupes localement compacts I. Ann. Sci. Ec. Norm. Super., IV. Ser. 7, 273–310 (1974)

    Google Scholar 

  3. Brunel, A., Revuz, D.: Marches de Harris sur les groupes localement compacts II. Bull. Soc. Math. France 104, 3–31 (1976)

    Google Scholar 

  4. Brunel, A., Revuz, D.: Marches de Harris sur les groupes localement compacts V. Ann. Inst. Herri Poincare, Nouv. Ser., Sect. B 15, 205–234 (1979)

    Google Scholar 

  5. Itô, K., McKean, H.P., Jr.: Potentials and the random walk. Ill. J. Math. 4, 119–132 (1960)

    Google Scholar 

  6. Kesten, H., Spitzer, F.: Random walk on countably infinite Abelian groups. Acta Math. 114, 237–265 (1965)

    Google Scholar 

  7. Neveu, J.: Potential Markovien recurrent des chaines de Harris. Ann. Inst. Fourier 22, 85–130 (1972)

    Google Scholar 

  8. Port, S.C., Stone, C.J.: Potential theory of random walks on Abelian groups. Acta Math. 122, 19–114 (1969)

    Google Scholar 

  9. Revuz, D.: Markov chains. Amsterdam Oxford: North Holland 1975

    Google Scholar 

  10. Spitzer, F.: A Tauberian theorem and its probability interpretation. Trans. Am. Math. Soc. 94, 150–169 (1960)

    Google Scholar 

  11. Spitzer, F.: Principles of random walk. Second edn. New York Heidelberg Berlin: Springer 1976

    Google Scholar 

Download references

Author information

Author notes
  1. Neal Madras

    Present address: Department of Mathematics, York University, 4700 Keele Street, M3J 1P3, Downsview, Ontario, Canada

Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, 251 Mercer Street, 10012, New York, New York, USA

    Neal Madras

Authors
  1. Neal Madras
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Madras, N. Random walks with killing. Probab. Th. Rel. Fields 80, 581–600 (1989). https://doi.org/10.1007/BF00318907

Download citation

  • Received: 17 February 1986

  • Revised: 20 July 1988

  • Issue Date: February 1989

  • DOI: https://doi.org/10.1007/BF00318907

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
  • Random Walk
  • Probability Theory
  • Statistical Theory
  • Walk Step
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