Home | Contents | Submissions, editors, etc. | Login | Search | EJP
 Electronic Communications in Probability > Vol. 15(2010) > Paper 17 open journal systems 


Green functions and Martin compactification for killed random walks related to SU(3)

Kilian Raschel, Universite Pierre et Marie Curie


Abstract
We consider the random walks killed at the boundary of the quarter plane, with homogeneous non-zero jump probabilities to the eight nearest neighbors and drift zero in the interior, and which admit a positive harmonic polynomial of degree three. For these processes, we find the asymptotic of the Green functions along all infinite paths of states, and from this we deduce that the Martin compactification is the one-point compactification.


Full text: PDF

Pages: 176-190

Published on: May 27, 2010
Bibliography
  1. Biane, P. Quantum random walk on the dual of SU(n). Probab. Theory Related Fields 89 (1991), no. 1, 117--129. Math. Review 93a:46119
  2. Biane, P. Équation de Choquet-Deny sur le dual d'un groupe compact. (French) Probab. Theory Related Fields 94 (1992), no. 1, 39--51. Math. Review 94a:46091
  3. Biane, P. Minuscule weights and random walks on lattices. Quantum probability & related topics, 51--65, QP-PQ, VII, World Sci. Publ., River Edge, NJ, 1992. Math. Review 94c:46129
  4. Collins, B. Martin boundary theory of some quantum random walks. Ann. Inst. H. Poincaré Probab. Statist. 40 (2004), no. 3, 367--384. Math. Review 2005m:31021
  5. Doob, J. L.; Snell, J. L.; Williamson, R. E. Application of boundary theory to sums of independent random variables. 1960 Contributions to probability and statistics pp. 182--197 Stanford Univ. Press, Stanford, Calif. Math. Review MR0120667
  6. Dynkin, E. B. The boundary theory of Markov processes (discrete case). (Russian) Uspehi Mat. Nauk 24 1969 no. 2 (146) 3--42. Math. Review MR0245096
  7. Fayolle, G.; Iasnogorodski, R.; Malyshev, V. Random walks in the quarter-plane. Algebraic methods, boundary value problems and applications. Applications of Mathematics (New York), 40. Springer-Verlag, Berlin, 1999. xvi+156 pp. ISBN: 3-540-65047-4 Math. Review 2000g:60002
  8. Ignatiouk-Robert, I. Martin boundary of a killed random walk on Z_+^d, preprint : arXiv, 2009
  9. Jones G. A.; Singerman D., Complex functions, Cambridge Univ. Press, Cambridge, 1987. Math. Review 89b:30001
  10. Kurkova I.; Raschel, K. Random walks in Z_+^2 with non-zero drift absorbed at the axes, to appear in Bulletin de la Société Mathématique de France, preprint : arXiv, 2009
  11. Ney, P.; Spitzer, F. The Martin boundary for random walk. Trans. Amer. Math. Soc. 121 1966 116--132. Math. Review MR0195151
  12. Picardello M. A.; Woess W. Martin boundaries of Cartesian products of Markov chains, Nagoya Math. J. 128 (1992), 153–169. Math. Review 94a:60109
  13. Raschel, K. Random walks in the quarter plane absorbed at the boundary : exact and asymptotic, preprint : arXiv, 2009
  14. Spitzer, F. Principles of random walk. The University Series in Higher Mathematics D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London 1964 xi+406 pp. Math. Review MR0171290
















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | EJP

Electronic Communications in Probability. ISSN: 1083-589X