## Illinois Journal of Mathematics

### Stable semigroups on homogeneous trees and hyperbolic spaces

Andrzej Stós

#### Abstract

We prove the kernel estimates for subordinated semigroups on homogeneous trees. We study the long time propagation problem. We exploit this to show exit time estimates for large balls in an abstract setting of metric measure spaces. Finally, we give estimates for the Poisson kernel of a ball.

#### Article information

Source
Illinois J. Math., Volume 55, Number 4 (2011), 1437-1454.

Dates
First available in Project Euclid: 12 July 2013

https://projecteuclid.org/euclid.ijm/1373636692

Digital Object Identifier
doi:10.1215/ijm/1373636692

Mathematical Reviews number (MathSciNet)
MR3082877

Zentralblatt MATH identifier
1276.60082

#### Citation

Stós, Andrzej. Stable semigroups on homogeneous trees and hyperbolic spaces. Illinois J. Math. 55 (2011), no. 4, 1437--1454. doi:10.1215/ijm/1373636692. https://projecteuclid.org/euclid.ijm/1373636692

#### References

• J.-P. Anker and L. Ji, Heat kernel and Green function estimates on noncompact symmetric spaces, Geom. Funct. Anal. 9 (1999), 1035–1091.
• J.-P. Anker and A. G. Setti, Asymptotic finite propagation speed for heat diffusion on certain Riemannian manifolds, J. Funct. Anal. 103 (1992), 50–61.
• M. Babillot, A probabilistic approach to heat diffusion on symmetric spaces, J. Theoret. Probab. 7 (1994), 599–607.
• M. T. Barlow, Diffusion on fractals, Lectures on probability theory and statistics, Ecole d'Ete de Probabilites de Saint-Flour XXV–-1995, Lecture Notes in Mathematics, vol. 1690, Springer, New York, 1999, pp. 1–121.
• J. Bertoin, Lévy processes, Cambridge University Press, Cambridge, 1996.
• K. Bogdan, A. Stós and P. Sztonyk, Harnack inequality for stable processes on $d$-sets, Studia Math. 158 (2003), 163–198.
• K. L. Chung and Z. Zhao, From Brownian motion to Schrödinger's equation, Springer-Verlag, New York, 1995.
• M. Cowling, S. Meda and A. G. Setti, Estimates for functions of the Laplace operator on homogeneous trees, Trans. Amer. Math. Soc. 352 (2000), 4271–4293.
• E. B. Davies, Heat kernels and spectral theory, Cambridge University Press, Cambridge, 1989.
• A. Figà Talamanca and C. Nebia, Harmonic analysis and representation theory for groups acting on homogeneous trees, London Math. Soc. Lecture Note Ser., vol. 162, Cambridge University Press, Cambridge, 1991.
• A. Figà Talamanca and M. Picardello, Harmonic analysis on free groups, Lecture Notes in Pure and Applied Mathematics, vol. 87, Marcel Dekker, New York, 1983.
• R. K. Getoor, Infinitely divisible probabilities on the hyperbolic plane, Pacific J. Math. 11 (1961), 1287–1308.
• P. Graczyk and T. Jakubowski, Exit times and Poisson kernels of the Ornstein–Uhlenbeck diffusion, Stoch. Models 24 (2008), 314–337.
• P. Graczyk and A. Stós, Transition density estimates for stable processes on symmetric spaces, Pacific J. Math. 217 (2004), 87–100.
• I. S. Gradstein and I. M. Ryzhik, Table of integrals, series and products, 6th ed., Academic Press, London, 2000.
• A. Grigoryan, Heat kernel and function theory on metric measure spaces, Heat kernels and analysis on manifolds, graphs, and metric spaces, Contemp. Math., vol. 338, American Mathematical Society, Providence, 2003, pp. 143–172.
• J. Hawkes, A lower Lipschitz condition for the stable subordinator, Z. Wahrsch. verw. Gebiete 17 (1971), 23–32.
• N. Ikeda and S. Watanabe, On some relations between the harmonic measure and the Lévy measure for a certain class of Markov processes, J. Math. Kyoto Univ. 2 (1962), 79–95.
• T. Jakubowski, The estimates of the mean first exit time from a ball for the $\alpha$-stable Ornstein–Uhlenbeck processes, Stochastic Process. Appl. 117 (2007), no. 10, 1540–1560.
• G. Medolla and A. G. Setti, Long time heat diffusion on homogeneous trees, Proc. Amer. Math. Soc. 128 (1999), 1733–1742.
• F. Olver, Asymptotics and special functions, Academic Press, New York, 1974.
• K. Pietruska-Pałuba, On function spaces related to fractional diffusions on $d$-sets, Stochastics Stochastics Rep. 70 (2000), 153–164.
• W. Woess, Heat diffusion on homogeneous trees (note on a paper by Medolla and Setti), Boll. Unione Mat. Ital. (8) 4-B (2001), 703–709.