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

Martin boundary of brownian motion on Gromov hyperbolic metric graphs

Soonki Hong 1,, and  Seonhee Lim 1,2,

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  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 $.

Keywords: General theory of random and stochastic dynamical systems, Dirichlet forms, Martin boundary theory.
Mathematics Subject Classification: Primary: 37H05; Secondary: 31C25, 31C35.
Citation: Soonki Hong, Seonhee Lim. Martin boundary of brownian motion on Gromov hyperbolic metric graphs. Discrete & 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.  Google Scholar

[2]

A. Ancona, Negatively curved manifolds, elliptic operators and the Martin boundary, Ann. of Math., 125 (1987), 495-536.  doi: 10.2307/1971409.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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M. Bonk and O. Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal., 10 (2000), 266-306.  doi: 10.1007/s000390050009.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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M. Brin and Y. Kifer, Brownian motion, harmonic functions and hyperbolicity for Euclidean complexes, Math. Z., 237 (2001), 421-468.  doi: 10.1007/PL00004875.  Google Scholar

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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.  Google Scholar

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J. Eells and B. Fuglede, Harmonic Maps, Between Riemannian Polyhedra, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 142 2001.  Google Scholar

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M. Fukushima, Dirichlet Forms and Markov Processes, North-Holland, Amsterdam and Tokyo, 1980.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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V. Kostrykin, J. Potthofff and R. Schrader, Brownian motions on metric graphs, J. Math. Phys., 53 (2012), 36 pp. doi: 10.1063/1.4714661.  Google Scholar

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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.  Google Scholar

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F. Ledrappier and S. Lim, Local limit theorem in negative curvature, to appear Duke Mathematics Journal, arXiv: 1503.04156. Google Scholar

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D. Lenz, P Stollmann and I. Veselić, The Allegretto-Piepenbrink theorem for strongly local Dirichlet forms, Documenta Math., 14 (2009), 167-189.   Google Scholar

[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.  Google Scholar

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R. McOwen, Partial Differential Equations: Methods and Applications, Prentice Hall, Upper Saddle River, NJ, 1996. Google Scholar

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J. R. Munkres, Topology, Second edition, Prentice Hall, Inc., Upper Saddle River, NJ, 2000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

M. L. Silverstein, Symmetric Markov Processes, Lecture Notes in Mathematics No. 426, Springer-Verlag, Berlin-New York, 1974 doi: 10.1007/BFb0073683.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

K- T Sturm, Analysis on local Dirichlet spaces-III. The parabolic Harnack inequality, J. Math. Pures Appl., 75 (1996), 273-297.   Google Scholar

[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.  Google Scholar

[36]

D. Sullivan, Related aspects of positivity in Riemannian geometry, J. Differential Geom., 25 (1987), 327-351.  doi: 10.4310/jdg/1214440979.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

A. Ancona, Negatively curved manifolds, elliptic operators and the Martin boundary, Ann. of Math., 125 (1987), 495-536.  doi: 10.2307/1971409.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

M. Bonk and O. Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal., 10 (2000), 266-306.  doi: 10.1007/s000390050009.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. Eells and B. Fuglede, Harmonic Maps, Between Riemannian Polyhedra, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 142 2001.  Google Scholar

[12]

M. Fukushima, Dirichlet Forms and Markov Processes, North-Holland, Amsterdam and Tokyo, 1980.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

M. L. Silverstein, Symmetric Markov Processes, Lecture Notes in Mathematics No. 426, Springer-Verlag, Berlin-New York, 1974 doi: 10.1007/BFb0073683.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

K- T Sturm, Analysis on local Dirichlet spaces-III. The parabolic Harnack inequality, J. Math. Pures Appl., 75 (1996), 273-297.   Google Scholar

[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.  Google Scholar

[36]

D. Sullivan, Related aspects of positivity in Riemannian geometry, J. Differential Geom., 25 (1987), 327-351.  doi: 10.4310/jdg/1214440979.  Google Scholar

[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.  Google Scholar

Figure 1.  Tree approximation
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Figure 2.  Ancona-Gouëzel inequality
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Figure 3.  Strong Ancona-Gouëzel inequality
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Figure 4.   
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