Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-21T09:47:42.501Z Has data issue: false hasContentIssue false
  • English
  • Français

Unbounded Laplacians on Graphs: Basic Spectral Properties andthe Heat Equation

Published online by Cambridge University Press:  12 May 2010

M. Keller  and
D. Lenz
Get access

Abstract

We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We focus onphenomena related to unboundedness of the Laplacians. This includes (failure of) essentialselfadjointness, absence of essential spectrum and stochastic incompleteness.

Keywords

Dirichlet formsgraphs, essential self adjointnessessential spectrumisoperimetric inequalitiesstochastic completeness
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 4: Spectral problems. Issue dedicated to the memory of M. Birman , 2010 , pp. 198 - 224
Copyright
© EDP Sciences, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Beurling, A., Deny, J.. Espaces de Dirichlet. I. Le cas élémentaire . Acta Math., 99 (1958), 203–224.CrossRefGoogle Scholar
Beurling, A., Deny, J.. Dirichlet spaces . Proc. Nat. Acad. Sci. U.S.A., 45 (1959), 208–215.CrossRefGoogle Scholar
N. Bouleau, F. Hirsch. Dirichlet forms and analysis on Wiener space. Volume 14 ofde Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 1991.
F. R. K. Chung. Spectral Graph Theory. CBMS Regional Conference Series in Mathematics, 92, American Mathematical Society, Providence, RI, 1997.
Chung, F. R. K., Grigoryan, A., Yau, S.-T.. Higher eigenvalues and isoperimetric inequalities on Riemannian manifolds and graphs . Comm. Anal. Geom., 8 (2000), No. 5, 969–1026.CrossRefGoogle Scholar
Y. Colin de Verdière. Spectres de graphes. Soc. Math. France, Paris, 1998.
E. B. Davies. Heat kernels and spectral theory. Cambridge University press, Cambridge, 1989.
E. B. Davies. Linear operators and their spectra. Cambridge Studies in Advanced Mathematics, 106. Cambridge University Press, Cambridge, 2007.
Dodziuk, J.. Difference Equations, isoperimetric inequality and transience of certain random walks . Trans. Amer. Math. Soc., 284 (1984), No. 2, 787–794.CrossRefGoogle Scholar
J. Dodziuk. Elliptic operators on infinite graphs. Analysis, geometry and topology of elliptic operators, 353–368, World Sci. Publ., Hackensack, NJ, 2006.
J. Dodziuk, W. S. Kendall. Combinatorial Laplacians and isoperimetric inequality. From local times to global geometry, control and physics (Coventry, 1984/85), 68–74, Pitman Res. Notes Math. Ser., 150, Longman Sci. Tech., Harlow, 1986.
J. Dodziuk, V. Matthai. Kato’s inequality and asymptotic spectral properties for discrete magnetic Laplacians. The ubiquitous heat kernel, 69–81, Contemp. Math., 398, Amer. Math. Soc., Providence, RI, 2006.
Feller, W.. On boundaries and lateral conditions for the Kolmogorov differential equations . Ann. of Math. (2), 65 (1957), 527–570.CrossRefGoogle Scholar
Fujiwara, K.. Laplacians on rapidly branching trees . Duke Math Jour., 83 (1996), No. 1, 191-202.CrossRefGoogle Scholar
M. Fukushima, Y. Oshima, M.Takeda. Dirichlet forms and symmetric Markov processes. de Gruyter Studies in Mathematics, 19. Walter de Gruyter & Co., Berlin, 1994.
Grigor’yan, A.. Analytic and geometric background of reccurrence and non-explosion of the brownian motion on riemannian manifolds . Bull. Am. Math. Soc., 36 (1999), No. 2, 135–249.CrossRefGoogle Scholar
S. Haeseler, M. Keller, Generalized solutions and spectrum for Dirichlet forms on graphs, preprint 2010, arXiv:1002.1040.
Häggström, O., Jonasson, J., Lyons, R.. Explicit isoperimetric constants and phase transitions in the random-cluster model . Ann. Probab., 30 (2002), No. 1, 443–473.Google Scholar
Higuchi, Y., Shirai, T.. Isoperimetric constants of (d,f)-regular planar graphs . Interdiscip. Inform. Sci., 9 (2003), No. 2, 221–228.Google Scholar
Jorgensen, P. E. T.. Essential selfadjointness of the graph-Laplacian . J. Math. Phys., 49 (2008), No. 7, 073510.CrossRefGoogle Scholar
Keller, M.. The essential spectrum of Laplacians on rapidly branching tesselations . Math. Ann., 346 (2010), No. 1, 51–66.CrossRefGoogle Scholar
M. Keller, D. Lenz. Dirichlet forms and stochastic completeness of graphs and subgraphs. preprint 2009, arXiv:0904.2985.
M. Keller, N. Peyerimhoff. Cheeger constants, growth and spectrum of locally tessellating planar graphs. to appear in Math. Z., arXiv:0903.4793.
Mohar, B.. Light structures in infinite planar graphs without the strong isoperimetric property . Trans. Amer. Math. Soc., 354 (2002), No. 8, 3059–3074.CrossRefGoogle Scholar
Z.-M. Ma and M. Röckner. Introduction to the theory of (non-symmetric) Dirichlet forms. Springer-Verlag, Berlin, 1992.
Metzger, B., Stollmann, P.. Heat kernel estimates on weighted graphs . Bull. London Math. Soc., 32 (2000), No. 4, 477–483.CrossRefGoogle Scholar
Reuter, G. E. H.. Denumerable Markov processes and the associated contraction semigroups on l . Acta Math., 97 (1957), 1–46.CrossRefGoogle Scholar
Sturm, K.-T.. textitAnalysis on local Dirichlet spaces. I: Recurrence, conservativeness and L p -Liouville properties. J. Reine Angew. Math., 456 (1994), No. 173–196. Google Scholar
Stollmann, P.. A convergence theorem for Dirichlet forms with applications to boundary value problems with varying domains . Math. Z., 219 (1995), No. 2, 275–287.CrossRefGoogle Scholar
Stollmann, P., Voigt, J.. Perturbation of Dirichlet forms by measures . Potential Anal. 5 (1996), No. 2, 109–138.CrossRefGoogle Scholar
Urakawa, H.. The spectrum of an infinite graph . Can. J. Math., 52 (2000), No. 5, 1057–1084.CrossRefGoogle Scholar
A. Weber. Analysis of the physical Laplacian and the heat flow on a locally finite graph. Preprint 2008, arXiv:0801.0812.
R. K. Wojciechowski. Stochastic completeness of graphs, PhD thesis, 2007. arXiv:0712.1570v2.
Wojciechowski, R. K.. Heat kernel and essential spectrum of infinite graphs . Indiana Univ. Math. J., 58 (2009), No. 3, 1419–1441.CrossRefGoogle Scholar
R. K. Wojciechowski. Stochastically Incomplete Manifolds and Graphs. Preprint 2009, arXiv:0910.5636.