no. 4
Brownian motion on treebolic space: positive harmonic functions
[Mouvement Brownien sur l’espace arbolique : fonctions harmoniques positives]
Bendikov, Alexander1 ; Saloff-Coste, Laurent2 ; Salvatori, Maura3 ; Woess, Wolfgang4
1 Insitute of Mathematics Wroclaw University Pl. Grundwaldzki 2/4 50-384 Wroclaw, Poland
2 Department of Mathematics Cornell University Ithaca, NY 14853, USA
3 Dipartimento di Matematica Università di Milano Via Saldini 50 20133 Milano, Italy
4 Institut für Diskrete Mathematik Technische Universität Graz Steyrergasse 30 A-8010 Graz, Austria
Annales de l'Institut Fourier, Tome 66 (2016) no. 4, pp. 1691-1731.

Ce travail est dedié à une étude de la théorie du potentiel sur l’espace arbolique, i.e., le produit horcyclique d’un ârbre régulier avec le demi-plan hyperbolique supérieur. En se basant sur l’analyse sur les complexes à bandes Riemanniennes développée par les auteurs, on considère une famille de Laplaciens avec deux paramètres concernant la dérive verticale. On examine les fonctions harmoniques associées à ces Laplaciens.

This paper studies potential theory on treebolic space, that is, the horocyclic product of a regular tree and hyperbolic upper half plane. Relying on the analysis on strip complexes developed by the authors, a family of Laplacians with “vertical drift” parameters is considered. We investigate the positive harmonic functions associated with those Laplacians.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3048
Classification : 31C05, 60J50, 53C23, 05C05
Keywords: Tree, hyperbolic plane, horocyclic product, quantum complex, Laplacian, positive harmonic functions
Mot clés : Arbre, plan hyperbolique, produit horocyclique, complexe quantique, Laplacien, fonctions harmoniques positives

Bendikov, Alexander 1 ; Saloff-Coste, Laurent 2 ; Salvatori, Maura 3 ; Woess, Wolfgang 4

1 Insitute of Mathematics Wroclaw University Pl. Grundwaldzki 2/4 50-384 Wroclaw, Poland
2 Department of Mathematics Cornell University Ithaca, NY 14853, USA
3 Dipartimento di Matematica Università di Milano Via Saldini 50 20133 Milano, Italy
4 Institut für Diskrete Mathematik Technische Universität Graz Steyrergasse 30 A-8010 Graz, Austria 
@article{AIF_2016__66_4_1691_0,
     author = {Bendikov, Alexander and Saloff-Coste, Laurent and Salvatori, Maura and Woess, Wolfgang},
     title = {Brownian motion on treebolic space: positive harmonic functions},
     journal = {Annales de l'Institut Fourier},
     pages = {1691--1731},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {66},
     number = {4},
     year = {2016},
     doi = {10.5802/aif.3048},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3048/}
}
TY  - JOUR
AU  - Bendikov, Alexander
AU  - Saloff-Coste, Laurent
AU  - Salvatori, Maura
AU  - Woess, Wolfgang
TI  - Brownian motion on treebolic space: positive harmonic functions
JO  - Annales de l'Institut Fourier
PY  - 2016
SP  - 1691
EP  - 1731
VL  - 66
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3048/
DO  - 10.5802/aif.3048
LA  - en
ID  - AIF_2016__66_4_1691_0
ER  - 
%0 Journal Article
%A Bendikov, Alexander
%A Saloff-Coste, Laurent
%A Salvatori, Maura
%A Woess, Wolfgang
%T Brownian motion on treebolic space: positive harmonic functions
%J Annales de l'Institut Fourier
%D 2016
%P 1691-1731
%V 66
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3048/
%R 10.5802/aif.3048
%G en
%F AIF_2016__66_4_1691_0
Bendikov, Alexander; Saloff-Coste, Laurent; Salvatori, Maura; Woess, Wolfgang. Brownian motion on treebolic space: positive harmonic functions. Annales de l'Institut Fourier, Tome 66 (2016) no. 4, pp. 1691-1731. doi : 10.5802/aif.3048. https://aif.centre-mersenne.org/articles/10.5802/aif.3048/

[1] Ancona, Alano Negatively curved manifolds, elliptic operators, and the Martin boundary, Ann. of Math. (2), Volume 125 (1987) no. 3, pp. 495-536 | DOI

[2] Bendikov, Alexander; Saloff-Coste, Laurent; Salvatori, Maura; Woess, Wolfgang The heat semigroup and Brownian motion on strip complexes, Adv. Math., Volume 226 (2011) no. 1, pp. 992-1055 | DOI

[3] Bendikov, Alexander; Saloff-Coste, Laurent; Salvatori, Maura; Woess, Wolfgang Brownian motion on treebolic space: escape to infinity, Rev. Mat. Iberoam., Volume 31 (2015) no. 3, pp. 935-976

[4] Blumenthal, R. M.; Getoor, R. K. Markov processes and potential theory, Pure and Applied Mathematics, Vol. 29, Academic Press, New York-London, 1968, x+313 pages

[5] Brelot, Marcel On topologies and boundaries in potential theory, Enlarged edition of a course of lectures delivered in 1966. Lecture Notes in Mathematics, Vol. 175, Springer-Verlag, Berlin-New York, 1971, vi+176 pages

[6] Brofferio, Sara; Salvatori, Maura; Woess, Wolfgang Brownian motion and harmonic functions on Sol (p,q), Int. Math. Res. Not. IMRN (2012) no. 22, pp. 5182-5218

[7] Brofferio, Sara; Woess, Wolfgang Green kernel estimates and the full Martin boundary for random walks on lamplighter groups and Diestel-Leader graphs, Ann. Inst. H. Poincaré Probab. Statist., Volume 41 (2005) no. 6, pp. 1101-1123 erratum in vol. 42 (2006), 773–774 | DOI

[8] Brofferio, Sara; Woess, Wolfgang Positive harmonic functions for semi-isotropic random walks on trees, lamplighter groups, and DL-graphs, Potential Anal., Volume 24 (2006) no. 3, pp. 245-265 | DOI

[9] Cartier, P. Fonctions harmoniques sur un arbre, Symposia Mathematica, Vol. IX (Convegno di Calcolo delle Probabilità, INDAM, Rome, 1971), Academic Press, London, 1972, pp. 203-270

[10] 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

[11] Constantinescu, Corneliu; Cornea, Aurel Potential theory on harmonic spaces, Springer-Verlag, New York-Heidelberg, 1972, viii+355 pages (With a preface by H. Bauer, Die Grundlehren der mathematischen Wissenschaften, Band 158)

[12] Cuno, Johannes; Sava-Huss, Ecaterina Random walks on Baumslag-Solitar groups (preprint, http://arxiv.org/abs/1510.00833)

[13] Diestel, Reinhard; Leader, Imre A conjecture concerning a limit of non-Cayley graphs, J. Algebraic Combin., Volume 14 (2001) no. 1, pp. 17-25 | DOI

[14] Dynkin, E. B. Markov processes. Vols. II, Die Grundlehren der Mathematischen Wissenschaften, Bände 121, 122, Academic Press Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965, viii+274 pages

[15] Eells, J.; Fuglede, B. Harmonic maps between Riemannian polyhedra, Cambridge Tracts in Mathematics, 142, Cambridge University Press, Cambridge, 2001, xii+296 pages (With a preface by M. Gromov)

[16] Élie, Laure Étude du renouvellement sur le groupe affine de la droite réelle, Ann. Sci. Univ. Clermont Math. (1977) no. 15, pp. 47-62

[17] Élie, Laure Fonctions harmoniques positives sur le groupe affine, Probability measures on groups (Proc. Fifth Conf., Oberwolfach, 1978) (Lecture Notes in Math.), Volume 706, Springer, Berlin, 1979, pp. 96-110

[18] Evans, Lawrence C. Partial differential equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998, xviii+662 pages

[19] Gouëzel, Sébastien Local limit theorem for symmetric random walks in Gromov-hyperbolic groups, J. Amer. Math. Soc., Volume 27 (2014) no. 3, pp. 893-928 | DOI

[20] Gouëzel, Sébastien Martin boundary of random walks with unbounded jumps in hyperbolic groups, Ann. Probab., Volume 43 (2015) no. 5, pp. 2374-2404 | DOI

[21] Helgason, Sigurdur Groups and geometric analysis, Pure and Applied Mathematics, 113, Academic Press, Inc., Orlando, FL, 1984, xix+654 pages (Integral geometry, invariant differential operators, and spherical functions)

[22] Karlsson, Anders; Ledrappier, François Linear drift and Poisson boundary for random walks, Pure Appl. Math. Q., Volume 3 (2007) no. 4, Special Issue: In honor of Grigory Margulis. Part 1, pp. 1027-1036 | DOI

[23] Karlsson, Anders; Ledrappier, François Propriété de Liouville et vitesse de fuite du mouvement brownien, C. R. Math. Acad. Sci. Paris, Volume 344 (2007) no. 11, pp. 685-690 | DOI

[24] Revuz, D. Markov chains, North-Holland Mathematical Library, 11, North-Holland Publishing Co., Amsterdam, 1984, xi+374 pages

[25] Woess, Wolfgang Lamplighters, Diestel-Leader graphs, random walks, and harmonic functions, Combin. Probab. Comput., Volume 14 (2005) no. 3, pp. 415-433 | DOI

[26] Woess, Wolfgang What is a horocyclic product, and how is it related to lamplighters?, Internat. Math. Nachrichten of the Austrian Math. Soc., Volume 224 (2013), pp. 1-27 (http://arxiv.org/abs/1401.1976)

Cité par Sources :