Electronic Communications in Probability

$t$-Martin boundary of reflected random walks on a half-space

Irina Ignatiouk-Robert

Full-text: Open access

Abstract

The $t$-Martin boundary of a random walk on a half-space with reflected boundary conditions is identified. It is shown in particular that the $t$-Martin boundary of such a random walk is not stable in the following sense: for different values of $t$, the $t$-Martin compactifications are not equivalent.

Article information

Source
Electron. Commun. Probab., Volume 15 (2010), paper no. 15, 149-161.

Dates
Accepted: 19 May 2010
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465243958

Digital Object Identifier
doi:10.1214/ECP.v15-1541

Mathematical Reviews number (MathSciNet)
MR2651547

Zentralblatt MATH identifier
1226.60104

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 31C35: Martin boundary theory [See also 60J50] 31C05: Harmonic, subharmonic, superharmonic functions 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 60J50: Boundary theory

Keywords
t-Martin boundary Markov chain stability

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Ignatiouk-Robert, Irina. $t$-Martin boundary of reflected random walks on a half-space. Electron. Commun. Probab. 15 (2010), paper no. 15, 149--161. doi:10.1214/ECP.v15-1541. https://projecteuclid.org/euclid.ecp/1465243958


Export citation

References

  • Billingsley, Patrick. Convergence of probability measures. John Wiley & Sons, Inc., New York-London-Sydney 1968 xii+253 pp.
  • Doob, J. L. Discrete potential theory and boundaries. J. Math. Mech. 8 1959 433–458; erratum 993.
  • Dupuis, Paul; Ellis, Richard S.; Weiss, Alan. Large deviations for Markov processes with discontinuous statistics. I. General upper bounds. Ann. Probab. 19 (1991), no. 3, 1280–1297.
  • Dupuis, Paul; Ellis, Richard S. Large deviations for Markov processes with discontinuous statistics. II. Random walks. Probab. Theory Related Fields 91 (1992), no. 2, 153–194.
  • Dupuis, Paul; Ellis, Richard S. The large deviation principle for a general class of queueing systems. I. Trans. Amer. Math. Soc. 347 (1995), no. 8, 2689–2751.
  • Hunt, G. A. Markoff chains and Martin boundaries. Illinois J. Math. 4 1960 313–340. (23 #A691)
  • Ignatiouk-Robert, Irina. Sample path large deviations and convergence parameters. Ann. Appl. Probab. 11 (2001), no. 4, 1292–1329.
  • Ignatiouk-Robert, Irina. Large deviations for processes with discontinuous statistics. Ann. Probab. 33 (2005), no. 4, 1479–1508.
  • Ignatiouk-Robert, Irina. On the spectrum of Markov semigroups via sample path large deviations. Probab. Theory Related Fields 134 (2006), no. 1, 44–80.
  • Ignatiouk-Robert, Irina. Martin boundary of a reflected random walk on a half-space. Probab. Theory Related Fields (2010), online, DOI: 10.1007/s00440-009-0228-4.
  • Ignatyuk, I. A.; Malyshev, V. A.; Shcherbakov, V. V. The influence of boundaries in problems on large deviations. (Russian) Uspekhi Mat. Nauk 49 (1994), no. 2(296), 43–102; translation in Russian Math. Surveys 49 (1994), no. 2, 41–99
  • Martin, Robert S. Minimal positive harmonic functions. Trans. Amer. Math. Soc. 49, (1941). 137–172.
  • Molčanov, S. A. Martin boundary of the direct product of Markov processes. (Russian) Uspehi Mat. Nauk 22 1967 no. 2 (134) 125–126.
  • Picardello, Massimo A.; Woess, Wolfgang. Martin boundaries of Cartesian products of Markov chains. Nagoya Math. J. 128 (1992), 153–169.
  • Picardello, Massimo; Woess, Wolfgang. Examples of stable Martin boundaries of Markov chains. Potential theory (Nagoya, 1990), 261–270, de Gruyter, Berlin, 1992.
  • Pruitt, William E. Eigenvalues of non-negative matrices. Ann. Math. Statist. 35 1964 1797–1800.
  • Rockafellar, R. Tyrrell. Convex analysis. Reprint of the 1970 original. Princeton Landmarks in Mathematics. Princeton Paperbacks. Princeton University Press, Princeton, NJ, 1997. xviii+451 pp. ISBN: 0-691-01586-4
  • Seneta, E. Nonnegative matrices and Markov chains. Second edition. Springer Series in Statistics. Springer-Verlag, New York, 1981. xiii+279 pp. ISBN: 0-387-90598-7
  • Woess, Wolfgang. Random walks on infinite graphs and groups. Cambridge Tracts in Mathematics, 138. Cambridge University Press, Cambridge, 2000. xii+334 pp. ISBN: 0-521-55292-3