
-
Previous Article
Cylinder absolute games on solenoids
- DCDS Home
- This Issue
-
Next Article
A Liouville theorem of parabolic Monge-AmpÈre equations in half-space
Martin boundary of brownian motion on Gromov hyperbolic metric graphs
1. | Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea |
2. | Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea |
Let $ \widetilde{X} $ be a locally finite Gromov hyperbolic graph whose Gromov boundary consists of infinitely many points and with a cocompact isometric action of a discrete group $ \Gamma $. We show the uniform Ancona inequality for the Brownian motion which implies that the $ \lambda $-Martin boundary coincides with the Gromov boundary for any $ \lambda \in [0, \lambda_0], $ in particular at the bottom of the spectrum $ \lambda_0 $.
References:
[1] |
S. Albeverio and M. Röckner,
Classical Dirichlet forms on topological spacesthe construction of an associated diffusion process, Probab. Th. Rel. Fields, 83 (1989), 405-434.
doi: 10.1007/BF00964372. |
[2] |
A. Ancona,
Negatively curved manifolds, elliptic operators and the Martin boundary, Ann. of Math., 125 (1987), 495-536.
doi: 10.2307/1971409. |
[3] |
A. F. Beardon, The Geometry of Discrete Groups, Graduate Texts in Mathematics, 91, Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4612-1146-4. |
[4] |
A. Bendikov, L. Saloff-Coste, M. Salvatori and W. Woess,
The heat semigroup and Brownian motion on strip complexes, Adv. in Math., 226 (2011), 992-1055.
doi: 10.1016/j.aim.2010.07.014. |
[5] |
M. Bonk and O. Schramm,
Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal., 10 (2000), 266-306.
doi: 10.1007/s000390050009. |
[6] |
P. Bougerol,
Théorème central limite local sur certains groupes de Lie, Ann. Sci. École Norm. Sup., 14 (1981), 403-432.
doi: 10.24033/asens.1412. |
[7] |
M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Fundamental Principles of Mathematical Sciences, 319. Springer-Verlag, Berlin, 1999.
doi: 10.1007/978-3-662-12494-9. |
[8] |
M. Brin and Y. Kifer,
Brownian motion, harmonic functions and hyperbolicity for Euclidean complexes, Math. Z., 237 (2001), 421-468.
doi: 10.1007/PL00004875. |
[9] |
S. Y. Cheng and S. T. Yao,
Differential equations on Riemannian manifolds and their geometric applications, Comm. on Pure Appl. Math., 28 (1975), 333-354.
doi: 10.1002/cpa.3160280303. |
[10] |
J. Dodziuk,
Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Univ. Math. J., 32 (1983), 703-716.
doi: 10.1512/iumj.1983.32.32046. |
[11] |
J. Eells and B. Fuglede, Harmonic Maps, Between Riemannian Polyhedra, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 142 2001. |
[12] |
M. Fukushima, Dirichlet Forms and Markov Processes, North-Holland, Amsterdam and Tokyo, 1980. |
[13] |
M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Process, De Gruyter Studies in Mathematics, 19 Walter de Gruyter, Berlin, 1994.
doi: 10.1515/9783110218091. |
[14] |
É. Ghys and P. de la Harpe, Sur Les Groupes Hyperboliques d'apr$\grave{e}$s Mikhael Gromov, Progress in Mathematics, 83, Birkhäuser Boston, Boston, MA, 1990.
doi: 10.1007/978-1-4684-9167-8. |
[15] |
S. Gouëzel,
Local limit theorem for symmetric random walks in Gromov-hyperbolic groups, J. Amer. Math. Soc., 27 (2014), 893-928.
doi: 10.1090/S0894-0347-2014-00788-8. |
[16] |
S. Gouëzel and S. P. Lalley,
Random walks on co-compact Fuchsian groups, Ann. Sci. École Norm. Sup., 46 (2013), 129-173.
doi: 10.24033/asens.2186. |
[17] |
A. Grigor'yan, Heat kernels and function theory on metric measure spaces, Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces, Contemp. Math., 338, Amer. Math. Soc., Providence, RI, 2003,143–172.
doi: 10.1090/conm/338/06073. |
[18] |
S. Haeseler and M. Keller, Generalized solutions and spectrum for Dirichlet forms on graphs, in Random Walks, Boundaries and Spectra, Progr. Probab., Birkhäuser/Springer Basel AG, Basel, 64 (2011), 181–199.
doi: 10.1007/978-3-0346-0244-0_10. |
[19] |
M. Keller and D. Lenz,
Dirichlet forms and stochastic completeness of graphs and subgraphs, J. Reine. Angew. Math., 666 (2012), 189-223.
doi: 10.1515/CRELLE.2011.122. |
[20] |
V. Kostrykin, J. Potthofff and R. Schrader, Brownian motions on metric graphs, J. Math. Phys., 53 (2012), 36 pp.
doi: 10.1063/1.4714661. |
[21] |
V. Kostrykin and R. Schrader, Laplacians on metric graphs: Eigenvalues, resolvents and semigroups, in Quantum Graphs and Their Applications, (edited by G. Berkolaiko, R. Carlson, S. A. Fulling, and P. Kuchment), Contemp. Math., Amer. Math. Soc., Providence, RI, 415 (2006), 201–225.
doi: 10.1090/conm/415. |
[22] |
F. Ledrappier and S. Lim, Local limit theorem in negative curvature, to appear Duke Mathematics Journal, arXiv: 1503.04156. Google Scholar |
[23] |
D. Lenz, P Stollmann and I. Veselić,
The Allegretto-Piepenbrink theorem for strongly local Dirichlet forms, Documenta Math., 14 (2009), 167-189.
|
[24] |
T. Lyons and D. Sullivan,
Function theory, random paths and covering spaces, J. Differential Geom., 19 (1984), 299-323.
doi: 10.4310/jdg/1214438681. |
[25] |
R. McOwen, Partial Differential Equations: Methods and Applications, Prentice Hall, Upper Saddle River, NJ, 1996. Google Scholar |
[26] |
J. R. Munkres, Topology, Second edition, Prentice Hall, Inc., Upper Saddle River, NJ, 2000. |
[27] |
M. Pivarski and L. Saloff-Coste, Small time heat kernel behavior on Riemannian complexes, New York J. Math., 14 (2008), 459–494, http://nyjm.albany.edu/j/2008/14_459.html. |
[28] |
L. Saloff-Coste and W. Woess,
Transition operators on co-compact G-spaces, Rev. Mat. Iberoam., 22 (2006), 747-799.
doi: 10.4171/RMI/473. |
[29] |
L. Saloff-Coste and W. Woess, Computations of spectral radii on $\mathcal G$-spaces, in Spectral Analysis in Geometry and Number Theory (edited by M. Kotani, H. Nalto and T. Tate), Contemp. Math., 484 (2009), 195–218.
doi: 10.1090/conm/484/09476. |
[30] |
K. Schmüdegen, Unbounded Self-adjoint Operators on Hilbert Space, Graduate Texts in Mathematics, 265, Springer, Dordrecht, 2012.
doi: 10.1007/978-94-007-4753-1. |
[31] |
M. L. Silverstein, Symmetric Markov Processes, Lecture Notes in Mathematics No. 426, Springer-Verlag, Berlin-New York, 1974
doi: 10.1007/BFb0073683. |
[32] |
K- T Sturm,
Analysis on local Dirichlet spaces-I. Recurrence, conservativeness and $L^p$-Liouville properties., J. Reine Angew. Math., 456 (1994), 173-196.
doi: 10.1515/crll.1994.456.173. |
[33] |
K-T Sturm, Analysis on local Dirichlet spaces-II. Upper Gaussian estimates for the fundamental solutions of parabolic equations, Osaka J. Math., 32 (1995), 275–312. https://projecteuclid.org/euclid.ojm/1200786053. |
[34] |
K- T Sturm,
Analysis on local Dirichlet spaces-III. The parabolic Harnack inequality, J. Math. Pures Appl., 75 (1996), 273-297.
|
[35] |
K-T Sturm, Metric measure spaces with variable Ricci bounds and couplings of Brownian motions, in Festschrift Masatoshi Fukushima, (edited by Z.-Q. Chen, N. Jacob, M. Takeda and T. Uemura), Interdiscip. Math. Sci, World Sci., World Sci. Publ., Hackensack, NJ, 17 (2015), 553–575.
doi: 10.1142/9789814596534_0027. |
[36] |
D. Sullivan,
Related aspects of positivity in Riemannian geometry, J. Differential Geom., 25 (1987), 327-351.
doi: 10.4310/jdg/1214440979. |
[37] |
R. K. Wojciechowski,
Heat kernel and essential spectrum of infinite graphs, Indiana Univ. Math. J., 58 (2009), 1419-1441.
doi: 10.1512/iumj.2009.58.3575. |
show all references
References:
[1] |
S. Albeverio and M. Röckner,
Classical Dirichlet forms on topological spacesthe construction of an associated diffusion process, Probab. Th. Rel. Fields, 83 (1989), 405-434.
doi: 10.1007/BF00964372. |
[2] |
A. Ancona,
Negatively curved manifolds, elliptic operators and the Martin boundary, Ann. of Math., 125 (1987), 495-536.
doi: 10.2307/1971409. |
[3] |
A. F. Beardon, The Geometry of Discrete Groups, Graduate Texts in Mathematics, 91, Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4612-1146-4. |
[4] |
A. Bendikov, L. Saloff-Coste, M. Salvatori and W. Woess,
The heat semigroup and Brownian motion on strip complexes, Adv. in Math., 226 (2011), 992-1055.
doi: 10.1016/j.aim.2010.07.014. |
[5] |
M. Bonk and O. Schramm,
Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal., 10 (2000), 266-306.
doi: 10.1007/s000390050009. |
[6] |
P. Bougerol,
Théorème central limite local sur certains groupes de Lie, Ann. Sci. École Norm. Sup., 14 (1981), 403-432.
doi: 10.24033/asens.1412. |
[7] |
M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Fundamental Principles of Mathematical Sciences, 319. Springer-Verlag, Berlin, 1999.
doi: 10.1007/978-3-662-12494-9. |
[8] |
M. Brin and Y. Kifer,
Brownian motion, harmonic functions and hyperbolicity for Euclidean complexes, Math. Z., 237 (2001), 421-468.
doi: 10.1007/PL00004875. |
[9] |
S. Y. Cheng and S. T. Yao,
Differential equations on Riemannian manifolds and their geometric applications, Comm. on Pure Appl. Math., 28 (1975), 333-354.
doi: 10.1002/cpa.3160280303. |
[10] |
J. Dodziuk,
Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Univ. Math. J., 32 (1983), 703-716.
doi: 10.1512/iumj.1983.32.32046. |
[11] |
J. Eells and B. Fuglede, Harmonic Maps, Between Riemannian Polyhedra, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 142 2001. |
[12] |
M. Fukushima, Dirichlet Forms and Markov Processes, North-Holland, Amsterdam and Tokyo, 1980. |
[13] |
M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Process, De Gruyter Studies in Mathematics, 19 Walter de Gruyter, Berlin, 1994.
doi: 10.1515/9783110218091. |
[14] |
É. Ghys and P. de la Harpe, Sur Les Groupes Hyperboliques d'apr$\grave{e}$s Mikhael Gromov, Progress in Mathematics, 83, Birkhäuser Boston, Boston, MA, 1990.
doi: 10.1007/978-1-4684-9167-8. |
[15] |
S. Gouëzel,
Local limit theorem for symmetric random walks in Gromov-hyperbolic groups, J. Amer. Math. Soc., 27 (2014), 893-928.
doi: 10.1090/S0894-0347-2014-00788-8. |
[16] |
S. Gouëzel and S. P. Lalley,
Random walks on co-compact Fuchsian groups, Ann. Sci. École Norm. Sup., 46 (2013), 129-173.
doi: 10.24033/asens.2186. |
[17] |
A. Grigor'yan, Heat kernels and function theory on metric measure spaces, Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces, Contemp. Math., 338, Amer. Math. Soc., Providence, RI, 2003,143–172.
doi: 10.1090/conm/338/06073. |
[18] |
S. Haeseler and M. Keller, Generalized solutions and spectrum for Dirichlet forms on graphs, in Random Walks, Boundaries and Spectra, Progr. Probab., Birkhäuser/Springer Basel AG, Basel, 64 (2011), 181–199.
doi: 10.1007/978-3-0346-0244-0_10. |
[19] |
M. Keller and D. Lenz,
Dirichlet forms and stochastic completeness of graphs and subgraphs, J. Reine. Angew. Math., 666 (2012), 189-223.
doi: 10.1515/CRELLE.2011.122. |
[20] |
V. Kostrykin, J. Potthofff and R. Schrader, Brownian motions on metric graphs, J. Math. Phys., 53 (2012), 36 pp.
doi: 10.1063/1.4714661. |
[21] |
V. Kostrykin and R. Schrader, Laplacians on metric graphs: Eigenvalues, resolvents and semigroups, in Quantum Graphs and Their Applications, (edited by G. Berkolaiko, R. Carlson, S. A. Fulling, and P. Kuchment), Contemp. Math., Amer. Math. Soc., Providence, RI, 415 (2006), 201–225.
doi: 10.1090/conm/415. |
[22] |
F. Ledrappier and S. Lim, Local limit theorem in negative curvature, to appear Duke Mathematics Journal, arXiv: 1503.04156. Google Scholar |
[23] |
D. Lenz, P Stollmann and I. Veselić,
The Allegretto-Piepenbrink theorem for strongly local Dirichlet forms, Documenta Math., 14 (2009), 167-189.
|
[24] |
T. Lyons and D. Sullivan,
Function theory, random paths and covering spaces, J. Differential Geom., 19 (1984), 299-323.
doi: 10.4310/jdg/1214438681. |
[25] |
R. McOwen, Partial Differential Equations: Methods and Applications, Prentice Hall, Upper Saddle River, NJ, 1996. Google Scholar |
[26] |
J. R. Munkres, Topology, Second edition, Prentice Hall, Inc., Upper Saddle River, NJ, 2000. |
[27] |
M. Pivarski and L. Saloff-Coste, Small time heat kernel behavior on Riemannian complexes, New York J. Math., 14 (2008), 459–494, http://nyjm.albany.edu/j/2008/14_459.html. |
[28] |
L. Saloff-Coste and W. Woess,
Transition operators on co-compact G-spaces, Rev. Mat. Iberoam., 22 (2006), 747-799.
doi: 10.4171/RMI/473. |
[29] |
L. Saloff-Coste and W. Woess, Computations of spectral radii on $\mathcal G$-spaces, in Spectral Analysis in Geometry and Number Theory (edited by M. Kotani, H. Nalto and T. Tate), Contemp. Math., 484 (2009), 195–218.
doi: 10.1090/conm/484/09476. |
[30] |
K. Schmüdegen, Unbounded Self-adjoint Operators on Hilbert Space, Graduate Texts in Mathematics, 265, Springer, Dordrecht, 2012.
doi: 10.1007/978-94-007-4753-1. |
[31] |
M. L. Silverstein, Symmetric Markov Processes, Lecture Notes in Mathematics No. 426, Springer-Verlag, Berlin-New York, 1974
doi: 10.1007/BFb0073683. |
[32] |
K- T Sturm,
Analysis on local Dirichlet spaces-I. Recurrence, conservativeness and $L^p$-Liouville properties., J. Reine Angew. Math., 456 (1994), 173-196.
doi: 10.1515/crll.1994.456.173. |
[33] |
K-T Sturm, Analysis on local Dirichlet spaces-II. Upper Gaussian estimates for the fundamental solutions of parabolic equations, Osaka J. Math., 32 (1995), 275–312. https://projecteuclid.org/euclid.ojm/1200786053. |
[34] |
K- T Sturm,
Analysis on local Dirichlet spaces-III. The parabolic Harnack inequality, J. Math. Pures Appl., 75 (1996), 273-297.
|
[35] |
K-T Sturm, Metric measure spaces with variable Ricci bounds and couplings of Brownian motions, in Festschrift Masatoshi Fukushima, (edited by Z.-Q. Chen, N. Jacob, M. Takeda and T. Uemura), Interdiscip. Math. Sci, World Sci., World Sci. Publ., Hackensack, NJ, 17 (2015), 553–575.
doi: 10.1142/9789814596534_0027. |
[36] |
D. Sullivan,
Related aspects of positivity in Riemannian geometry, J. Differential Geom., 25 (1987), 327-351.
doi: 10.4310/jdg/1214440979. |
[37] |
R. K. Wojciechowski,
Heat kernel and essential spectrum of infinite graphs, Indiana Univ. Math. J., 58 (2009), 1419-1441.
doi: 10.1512/iumj.2009.58.3575. |



[1] |
Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020451 |
[2] |
Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024 |
[3] |
Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020284 |
[4] |
Nitha Niralda P C, Sunil Mathew. On properties of similarity boundary of attractors in product dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021004 |
[5] |
Editorial Office. Retraction: Xiao-Qian Jiang and Lun-Chuan Zhang, A pricing option approach based on backward stochastic differential equation theory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 969-969. doi: 10.3934/dcdss.2019065 |
[6] |
Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3171-3200. doi: 10.3934/dcds.2020054 |
[7] |
Mauricio Achigar. Extensions of expansive dynamical systems. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020399 |
[8] |
Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020458 |
[9] |
Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310 |
[10] |
Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, 2021, 14 (1) : 115-148. doi: 10.3934/krm.2020051 |
[11] |
The Editors. The 2019 Michael Brin Prize in Dynamical Systems. Journal of Modern Dynamics, 2020, 16: 349-350. doi: 10.3934/jmd.2020013 |
[12] |
Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276 |
[13] |
Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen, Chih-Her Chen. On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3291-3304. doi: 10.3934/dcds.2020127 |
[14] |
Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020048 |
[15] |
Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324 |
[16] |
Mikhail I. Belishev, Sergey A. Simonov. A canonical model of the one-dimensional dynamical Dirac system with boundary control. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021003 |
[17] |
Toshiko Ogiwara, Danielle Hilhorst, Hiroshi Matano. Convergence and structure theorems for order-preserving dynamical systems with mass conservation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3883-3907. doi: 10.3934/dcds.2020129 |
[18] |
Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331 |
[19] |
Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021012 |
[20] |
Stefan Siegmund, Petr Stehlík. Time scale-induced asynchronous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1011-1029. doi: 10.3934/dcdsb.2020151 |
2019 Impact Factor: 1.338
Tools
Article outline
Figures and Tables
[Back to Top]