Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation

M. Keller; D. Lenz

Mathematical Modelling of Natural Phenomena (2010)

  • Volume: 5, Issue: 4, page 198-224
  • ISSN: 0973-5348

Abstract

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

How to cite

top

Keller, M., and Lenz, D.. "Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation." Mathematical Modelling of Natural Phenomena 5.4 (2010): 198-224. <http://eudml.org/doc/197705>.

@article{Keller2010,
abstract = {We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We focus on phenomena related to unboundedness of the Laplacians. This includes (failure of) essential selfadjointness, absence of essential spectrum and stochastic incompleteness.},
author = {Keller, M., Lenz, D.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {Dirichlet forms; graphs, essential self adjointness; essential spectrum; isoperimetric inequalities; stochastic completeness; graphs; essential selfadjointness},
language = {eng},
month = {5},
number = {4},
pages = {198-224},
publisher = {EDP Sciences},
title = {Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation},
url = {http://eudml.org/doc/197705},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Keller, M.
AU - Lenz, D.
TI - Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/5//
PB - EDP Sciences
VL - 5
IS - 4
SP - 198
EP - 224
AB - We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We focus on phenomena related to unboundedness of the Laplacians. This includes (failure of) essential selfadjointness, absence of essential spectrum and stochastic incompleteness.
LA - eng
KW - Dirichlet forms; graphs, essential self adjointness; essential spectrum; isoperimetric inequalities; stochastic completeness; graphs; essential selfadjointness
UR - http://eudml.org/doc/197705
ER -

References

top
  1. A. Beurling, J. Deny. Espaces de Dirichlet. I. Le cas élémentaire. Acta Math., 99 (1958), 203–224. 
  2. A. Beurling, J. Deny. Dirichlet spaces. Proc. Nat. Acad. Sci. U.S.A., 45 (1959), 208–215. 
  3. N. Bouleau, F. Hirsch. Dirichlet forms and analysis on Wiener space. Volume 14 ofde Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 1991.  
  4. F. R. K. Chung. Spectral Graph Theory. CBMS Regional Conference Series in Mathematics, 92, American Mathematical Society, Providence, RI, 1997.  
  5. F. R. K. Chung, A. Grigoryan, S.-T. Yau. Higher eigenvalues and isoperimetric inequalities on Riemannian manifolds and graphs. Comm. Anal. Geom., 8 (2000), No. 5, 969–1026. 
  6. Y. Colin de Verdière. Spectres de graphes. Soc. Math. France, Paris, 1998.  
  7. E. B. Davies. Heat kernels and spectral theory. Cambridge University press, Cambridge, 1989.  
  8. E. B. Davies. Linear operators and their spectra. Cambridge Studies in Advanced Mathematics, 106. Cambridge University Press, Cambridge, 2007.  
  9. J. Dodziuk. Difference Equations, isoperimetric inequality and transience of certain random walks. Trans. Amer. Math. Soc., 284 (1984), No. 2, 787–794. 
  10. J. Dodziuk. Elliptic operators on infinite graphs. Analysis, geometry and topology of elliptic operators, 353–368, World Sci. Publ., Hackensack, NJ, 2006.  
  11. 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.  
  12. 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.  
  13. W. Feller. On boundaries and lateral conditions for the Kolmogorov differential equations. Ann. of Math. (2), 65 (1957), 527–570. 
  14. K. Fujiwara. Laplacians on rapidly branching trees. Duke Math Jour., 83 (1996), No. 1, 191-202. 
  15. M. Fukushima, Y. Oshima, M.Takeda. Dirichlet forms and symmetric Markov processes. de Gruyter Studies in Mathematics, 19. Walter de Gruyter & Co., Berlin, 1994.  
  16. A. Grigor’yan. 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. 
  17. S. Haeseler, M. Keller, Generalized solutions and spectrum for Dirichlet forms on graphs, preprint 2010, arXiv:1002.1040.  
  18. O. Häggström, J. Jonasson, R. Lyons. Explicit isoperimetric constants and phase transitions in the random-cluster model. Ann. Probab., 30 (2002), No. 1, 443–473. 
  19. Y. Higuchi, T. Shirai. Isoperimetric constants of (d,f)-regular planar graphs. Interdiscip. Inform. Sci., 9 (2003), No. 2, 221–228. 
  20. P. E. T. Jorgensen. Essential selfadjointness of the graph-Laplacian. J. Math. Phys., 49 (2008), No. 7, 073510. 
  21. M. Keller. The essential spectrum of Laplacians on rapidly branching tesselations. Math. Ann., 346 (2010), No. 1, 51–66. 
  22. M. Keller, D. Lenz. Dirichlet forms and stochastic completeness of graphs and subgraphs. preprint 2009, arXiv:0904.2985.  
  23. M. Keller, N. Peyerimhoff. Cheeger constants, growth and spectrum of locally tessellating planar graphs. to appear in Math. Z., arXiv:0903.4793.  
  24. B. Mohar. Light structures in infinite planar graphs without the strong isoperimetric property. Trans. Amer. Math. Soc., 354 (2002), No. 8, 3059–3074. 
  25. Z.-M. Ma and M. Röckner. Introduction to the theory of (non-symmetric) Dirichlet forms. Springer-Verlag, Berlin, 1992.  
  26. B. Metzger, P. Stollmann. Heat kernel estimates on weighted graphs. Bull. London Math. Soc., 32 (2000), No. 4, 477–483. 
  27. G. E. H. Reuter. Denumerable Markov processes and the associated contraction semigroups onl. Acta Math., 97 (1957), 1–46. 
  28. K.-T. Sturm. textitAnalysis on local Dirichlet spaces. I: Recurrence, conservativeness and Lp-Liouville properties. J. Reine Angew. Math., 456 (1994), No. 173–196.  
  29. P. Stollmann. A convergence theorem for Dirichlet forms with applications to boundary value problems with varying domains. Math. Z., 219 (1995), No. 2, 275–287. 
  30. P. Stollmann, J. Voigt. Perturbation of Dirichlet forms by measures. Potential Anal.5 (1996), No. 2, 109–138. 
  31. H. Urakawa. The spectrum of an infinite graph. Can. J. Math., 52 (2000), No. 5, 1057–1084. 
  32. A. Weber. Analysis of the physical Laplacian and the heat flow on a locally finite graph. Preprint 2008, arXiv:0801.0812.  
  33. R. K. Wojciechowski. Stochastic completeness of graphs, PhD thesis, 2007. arXiv:0712.1570v2.  
  34. R. K. Wojciechowski. Heat kernel and essential spectrum of infinite graphs. Indiana Univ. Math. J., 58 (2009), No. 3, 1419–1441. 
  35. R. K. Wojciechowski. Stochastically Incomplete Manifolds and Graphs. Preprint 2009, arXiv:0910.5636.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.