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.
Revised:
Accepted:
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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}, 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 TI - Brownian motion on treebolic space: positive harmonic functions JO - Annales de l'Institut Fourier PY - 2016 DA - 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/ UR - https://doi.org/10.5802/aif.3048 DO - 10.5802/aif.3048 LA - en ID - AIF_2016__66_4_1691_0 ER -
Bendikov, Alexander; Saloff-Coste, Laurent; Salvatori, Maura; Woess, Wolfgang. Brownian motion on treebolic space: positive harmonic functions. Annales de l'Institut Fourier, Volume 66 (2016) no. 4, pp. 1691-1731. doi : 10.5802/aif.3048. https://aif.centre-mersenne.org/articles/10.5802/aif.3048/
[1] Negatively curved manifolds, elliptic operators, and the Martin boundary, Ann. of Math. (2), Tome 125 (1987) no. 3, pp. 495-536 | Article
[2] The heat semigroup and Brownian motion on strip complexes, Adv. Math., Tome 226 (2011) no. 1, pp. 992-1055 | Article
[3] Brownian motion on treebolic space: escape to infinity, Rev. Mat. Iberoam., Tome 31 (2015) no. 3, pp. 935-976
[4] Markov processes and potential theory, Pure and Applied Mathematics, Vol. 29, Academic Press, New York-London, 1968, x+313 pages
[5] 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] Brownian motion and harmonic functions on , Int. Math. Res. Not. IMRN (2012) no. 22, pp. 5182-5218
[7] 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] 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] 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] 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] 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] Random walks on Baumslag-Solitar groups (preprint, http://arxiv.org/abs/1510.00833)
[13] A conjecture concerning a limit of non-Cayley graphs, J. Algebraic Combin., Tome 14 (2001) no. 1, pp. 17-25 | Article
[14] Markov processes. Vols. II, Die Grundlehren der Mathematischen Wissenschaften, Bände 121, Tome 122, Academic Press Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965, viii+274 pages
[15] Harmonic maps between Riemannian polyhedra, Cambridge Tracts in Mathematics, Tome 142, Cambridge University Press, Cambridge, 2001, xii+296 pages (With a preface by M. Gromov)
[16] Étude du renouvellement sur le groupe affine de la droite réelle, Ann. Sci. Univ. Clermont Math. (1977) no. 15, pp. 47-62
[17] Fonctions harmoniques positives sur le groupe affine, Probability measures on groups (Proc. Fifth Conf., Oberwolfach, 1978) (Lecture Notes in Math.) Tome 706, Springer, Berlin, 1979, pp. 96-110
[18] Partial differential equations, Graduate Studies in Mathematics, Tome 19, American Mathematical Society, Providence, RI, 1998, xviii+662 pages
[19] 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] Martin boundary of random walks with unbounded jumps in hyperbolic groups, Ann. Probab., Tome 43 (2015) no. 5, pp. 2374-2404 | Article
[21] Groups and geometric analysis, Pure and Applied Mathematics, Tome 113, Academic Press, Inc., Orlando, FL, 1984, xix+654 pages (Integral geometry, invariant differential operators, and spherical functions)
[22] 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] 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] Markov chains, North-Holland Mathematical Library, Tome 11, North-Holland Publishing Co., Amsterdam, 1984, xi+374 pages
[25] Lamplighters, Diestel-Leader graphs, random walks, and harmonic functions, Combin. Probab. Comput., Tome 14 (2005) no. 3, pp. 415-433 | Article
[26] 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)
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