Brownian motion on treebolic space: positive harmonic functions
Annales de l'Institut Fourier, Volume 66 (2016) no. 4, p. 1691-1731
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.
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.
Received : 2015-03-24
Revised : 2015-11-15
Accepted : 2015-12-21
Published online : 2016-07-28
Classification:  31C05,  60J50,  53C23,  05C05
Keywords: Tree, hyperbolic plane, horocyclic product, quantum complex, Laplacian, positive harmonic functions
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {4},
     year = {2016},
     pages = {1691-1731},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2016__66_4_1691_0}
}
Brownian motion on treebolic space: positive harmonic functions. Annales de l'Institut Fourier, Volume 66 (2016) no. 4, pp. 1691-1731. https://aif.centre-mersenne.org/item/AIF_2016__66_4_1691_0/

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

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

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

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

[5] Brelot, Marcel On topologies and boundaries in potential theory, Springer-Verlag, Berlin-New York, Enlarged edition of a course of lectures delivered in 1966. Lecture Notes in Mathematics, Vol. 175 (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., Tome 41 (2005) no. 6, pp. 1101-1123 (erratum in vol. 42 (2006), 773–774) | Article

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

[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), Tome 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., Tome 14 (2001) no. 1, pp. 17-25 | Article

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

[15] Eells, J.; Fuglede, B. Harmonic maps between Riemannian polyhedra, Cambridge University Press, Cambridge, Cambridge Tracts in Mathematics, Tome 142 (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), Springer, Berlin (Lecture Notes in Math.) Tome 706 (1979), pp. 96-110

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

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

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

[21] Helgason, Sigurdur Groups and geometric analysis, Academic Press, Inc., Orlando, FL, Pure and Applied Mathematics, Tome 113 (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., Tome 3 (2007) no. 4, Special Issue: In honor of Grigory Margulis. Part 1, pp. 1027-1036 | Article

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

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

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

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