Abstract
The harmonic measure ν on the boundary of the group Sol associated to a discrete random walk of law µ was described by Kaimanovich. We investigate when it is absolutely continuous or singular with respect to Lebesgue measure. By ratio entropy over speed, we show that any countable non-abelian subgroup admits a finite first moment non-degenerate μ with singular harmonic measure ν. On the other hand, we prove that some random walks with finitely supported step distribution admit a regular harmonic measure. Finally, we construct some exceptional random walks with arbitrarily small speed but singular harmonic measures. The two later results are obtained by comparison with Bernoulli convolutions, using results of Erdős and Solomyak.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
B. Bárány, M. Pollicott and K. Simon, Stationary measures for projective transformations: The Blackwell and Furstenberg measures, Journal of Statistical Physics 148 (2012), 393–421.
A. Bendikov, L. Saloff-Coste, M. Salvatori and W. Woess, Brownian motion on treebolic space: escape to infinity, arXiv:1212.6151v2 [math.PR]_23 Feb 2013.
D. Bertacchi, Random walks on Diestel-Leader graphs, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 71 (2001), 205–224.
J. Bourgain, Finitely supported measures on SL2(R) which are absolutely continuous at infinity, in Geometric Aspect of Functional Analysis, Lecture Notes in Mathematics, Vol. 2050, Springer-Verlag, Berlin-Heidelberg, 2012, pp. 133–141.
S. Brofferio, M. Salvatori and W. Woess, Brownian motion and harmonic functions on Sol(p, q), International Mathematics Research Notices 22 (2012), 5182–5218.
W. Donoghue, Distributions and Fourier Transforms, Pure and Applied Mathematics, Vol. 32, Academic Press, Amsterdam, 1969.
P. Erdös, On a family of symmetric Bernoulli convolutions, American Journal of Mathematics 61 (1939), 974–976.
P. Erdös, On the smoothness properties of a family of Bernoulli convolutions, American Journal of Mathematics 62 (1940), 180–186.
A. Eskin, D. Fisher and K. Whyte, Quasi-isometries and rigidity of solvable groups, Pure and Applied Mathematics Quarterly 3 (2007), 927–947.
A. Eskin, D. Fisher and K. Whyte, Coarse differentiation of quasi-isometries I: Spaces not quasi-isometric to Cayley graphs, Annals of Mathematics 176 (2012), 221–260.
H. Furstenberg, Random walks and discrete subgroups of Lie groups, in Advances in Probability and Related Topics, Vol. 1, Dekker, New York, 1971, pp. 1–63.
V. Gadre, Harmonic measures for distributions with finite support on the mapping class group are singular, Duke Mathematical Journal 163 (2014), 309–368.
J.-P. Kahane, Sur la distribution de certaines séries aléatoires, in Colloques Théorie des Nombres (Univ. Bordeaux, Bordeaux, 1969), Mémoires de la Société Mathématique de France, Vol. 25, Société Mathématique de France, Paris, 1971, pp. 119–122.
V. A. Kaimanovich, Poisson boundaries of random walks on discrete solvable groups, in Probability Measures on Groups, X (Oberwolfach, 1990), Plenum, New York, 1991, pp. 205–238.
V. A. Kaimanovich, The Poisson formula for groups with hyperbolic properties, Annals of Mathematics 152 (2000), 659–692.
V. A. Kaimanovich and V. Le Prince, Matrix random products with singular harmonic measure, Geometriae Dedicata 150 (2011) 257–279.
V. A. Kaimanovich and H. Masur, The Poisson boundary of the mapping class group, Inventiones mathematicae 125 (1996), 221–264.
V. A. Kaimanovich and A. M. Vershik, Random walks on discrete groups: boundary and entropy, Annals of Probability 11 (1983), 457–490.
V. A. Kaimanovich and W. Woess, Boundary and entropy of space homogeneous Markov chains, Annals of Probability 30 (2002), 323–363.
F. Ledrappier, Une relation entre entropie, dimension et exposant pour certaines marches aléatoires, Comtes Rendus de l’Académie des Sciences. Série I. Mathématique 296 (1983), 369–372.
F. Ledrappier, Some asymptotic properties of random walks on free groups, in Topics in Probability and Lie Groups: Boundary Theory, CRM Proceedings and Lecture Notes, Vol. 28, American Mathematical Society, Providence, RI, 2001, pp. 117–152.
A. Medina and P. Revoy, Lattices in symplectic Lie groups, Journal of Lie Theory 17 (2007), 27–39.
E. Molnar and J. Szirmai, Classification of Sol lattices, Geometriae Dedicata 161 (2012), 251–275.
Y. Peres, W. Schlag and B. Solomyak, Sixty years of Bernoulli convolutions, in Fractal Geometry and Stochastics. II (Greifswald/Koserow, 1998), Progress in Probability, Vol. 46, Birkhäuser, Basel, 2000, pp. 39–65.
Y. Peres and B. Solomyak, Absolute continuity of Bernoulli convolutions, a simple proof, Mathematical Research Letters 3 (1996), 231–239.
Y. Peres and B. Solomyak, Self-similar measures and intersections of Cantor sets, Transactions of the American Mathematical Society 350 (1998), 40650–4087.
Y. B. Pesin, Dimension Theory in Dynamical Systems, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997.
P. Shmerkin, On the exceptional set for absolute continuity of Bernoulli convolutions, Geometric and Functional Analysis 24 (2014), 946–958.
B. Solomyak, On the random series Σ ± λ n (an Erdős problem), Annals of Mathematics 142 (1995), 611–625.
M. Troyanov, L’horizon de SOL, Expositiones Mathematicae 16 (1998), 441–479.
A. Wintner, On convergent Poisson convolutions, American Journal of Mathematics 57 (1935), 827–838.
W. Woess, What is a horocyclic product, and how is it related to lamplighters?, Internationale Matematische Nachrichten 224 (2013), 1–27.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Brieussel, J., Tanaka, R. Discrete random walks on the group Sol. Isr. J. Math. 208, 291–321 (2015). https://doi.org/10.1007/s11856-015-1200-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-015-1200-x