August  2021, 41(8): 3725-3757. doi: 10.3934/dcds.2021014

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

* Corresponding author: Soonki Hong

Received  July 2020 Published  August 2021 Early access  January 2021

Fund Project: This article based on Soonki Hong's Ph.D thesis

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 $.

Citation: Soonki Hong, Seonhee Lim. Martin boundary of brownian motion on Gromov hyperbolic metric graphs. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3725-3757. doi: 10.3934/dcds.2021014
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. BendikovL. Saloff-CosteM. 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.

[23]

D. LenzP 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.

[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. BendikovL. Saloff-CosteM. 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.

[23]

D. LenzP 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.

[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.

Figure 1.  Tree approximation
Figure 2.  Ancona-Gouëzel inequality
Figure 3.  Strong Ancona-Gouëzel inequality
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