The Poisson boundary of random rational affinities
Annales de l'Institut Fourier, Volume 56 (2006) no. 2, pp. 499-515.

We prove that in order to describe the Poisson boundary of rational affinities, it is necessary and sufficient to consider the action on real and all p-adic fileds.

On prouve que pour décrire la frontière de Poisson des affinités à coefficients rationnels est nécessaire et suffisant de considérer l’action sur le corps réel et tous les corps p-adiques.

DOI: 10.5802/aif.2191
Classification: 60B99, 60J50, 43A05, 22E35
Keywords: Poisson boundary, random walks, affine group, rational numbers, $p$-adic numbers
Mot clés : frontière de Poisson, marches aléatoires, groupe affine, nombres rationnels, nombres $p$-adiques

Brofferio, Sara 1

1 Technische Universität Graz Institut für Mathematik C Steyergasse 30 A-8010 Graz (Austria)
@article{AIF_2006__56_2_499_0,
     author = {Brofferio, Sara},
     title = {The {Poisson} boundary of random rational affinities},
     journal = {Annales de l'Institut Fourier},
     pages = {499--515},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {56},
     number = {2},
     year = {2006},
     doi = {10.5802/aif.2191},
     mrnumber = {2226025},
     zbl = {1087.60011},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2191/}
}
TY  - JOUR
AU  - Brofferio, Sara
TI  - The Poisson boundary of random rational affinities
JO  - Annales de l'Institut Fourier
PY  - 2006
SP  - 499
EP  - 515
VL  - 56
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2191/
DO  - 10.5802/aif.2191
LA  - en
ID  - AIF_2006__56_2_499_0
ER  - 
%0 Journal Article
%A Brofferio, Sara
%T The Poisson boundary of random rational affinities
%J Annales de l'Institut Fourier
%D 2006
%P 499-515
%V 56
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2191/
%R 10.5802/aif.2191
%G en
%F AIF_2006__56_2_499_0
Brofferio, Sara. The Poisson boundary of random rational affinities. Annales de l'Institut Fourier, Volume 56 (2006) no. 2, pp. 499-515. doi : 10.5802/aif.2191. https://aif.centre-mersenne.org/articles/10.5802/aif.2191/

[1] Babillot, Martine; Bougerol, Philippe; Elie, Laure The random difference equation X n =A n X n-1 +B n in the critical case, Ann. Probab., Volume 25 (1997) no. 1, pp. 478-493 | DOI | MR | Zbl

[2] Bougerol, Philippe; Picard, Nico Strict stationarity of generalized autoregressive processes, Ann. Probab., Volume 20 (1992) no. 4, pp. 1714-1730 | DOI | MR | Zbl

[3] Brofferio, Sara Poisson boundary for finitely generated groups of rational affinities (Preprint)

[4] Brofferio, Sara Speed of locally contractive stochastic systems, Ann. Probab., Volume 31 (2003) no. 4, pp. 2040-2067 | DOI | MR | Zbl

[5] Cartwright, D. I.; Kaĭmanovich, V. A.; Woess, W. Random walks on the affine group of local fields and of homogeneous trees, Ann. Inst. Fourier (Grenoble), Volume 44 (1994) no. 4, pp. 1243-1288 | DOI | Numdam | MR | Zbl

[6] Cassels, J. W. S. Global fields, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 42-84 | MR

[7] Derriennic, Y. Entropie, théorèmes limite et marches aléatoires, Probability measures on groups, VIII (Oberwolfach, 1985) (Lecture Notes in Math.), Volume 1210, Springer, Berlin, 1986, pp. 241-284 | MR | Zbl

[8] Derriennic, Yves Quelques applications du théorème ergodique sous-additif, Conference on Random Walks (Kleebach, 1979) (French) (Astérisque), Volume 74, Soc. Math. France, Paris, 1980, p. 183-201, 4 | MR | Zbl

[9] Élie, Laure Noyaux potentiels associés aux marches aléatoires sur les espaces homogènes. Quelques exemples clefs dont le groupe affine, Théorie du potentiel (Orsay, 1983) (Lecture Notes in Math.), Volume 1096, Springer, Berlin, 1984, pp. 223-260 | MR | Zbl

[10] Furstenberg, Harry Boundary theory and stochastic processes on homogeneous spaces, Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972), Amer. Math. Soc., Providence, R.I., 1973, pp. 193-229 | MR | Zbl

[11] Kaĭmanovich, V. A.; Vershik, A. M. Random walks on discrete groups: boundary and entropy, Ann. Probab., Volume 11 (1983) no. 3, pp. 457-490 | DOI | MR | Zbl

[12] Kaimanovich, Vadim A. Poisson boundaries of random walks on discrete solvable groups, Probability measures on groups, X (Oberwolfach, 1990), Plenum, New York, 1991, pp. 205-238 | MR | Zbl

[13] Kaimanovich, Vadim A. The Poisson formula for groups with hyperbolic properties, Ann. of Math. (2), Volume 152 (2000) no. 3, pp. 659-692 | DOI | MR | Zbl

[14] Kesten, Harry Random difference equations and renewal theory for products of random matrices, Acta Math., Volume 131 (1973), pp. 207-248 | DOI | MR | Zbl

[15] Lang, Serge Introduction to diophantine approximations, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966 | MR | Zbl

[16] Vervaat, Wim On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables, Adv. in Appl. Probab., Volume 11 (1979) no. 4, pp. 750-783 | DOI | MR | Zbl

Cited by Sources: