Advertisement

Mathematische Annalen

, Volume 331, Issue 1, pp 21–39 | Cite as

Strongly subharmonic functions, graphs, and their asymptotic growth

  • Marco Rigoli
  • Maura Salvatori
  • Marco VignatiEmail author
Article

Abstract.

Let G be an infinite connected graph with uniformly bounded vertex degree, and let Δ denote the Laplace operator corresponding to the simple random walk on it. In this paper we obtain relations between the structure of the graph and the qualitative behaviour of the class of functions u satisfying Δub>0, namely we relate the asymptotic growth of the function u and that of the cardinality of balls in G.

Keywords

Random Walk Laplace Operator Connected Graph Qualitative Behaviour Vertex Degree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ancona, A.: Thèorie du potential sur les graphes et les variétés. Ècole de probabilité de St.Flour, Springer Lecture Notes in Mathematics, 1427, pp. 1–311Google Scholar
  2. 2.
    Bendito, E., Carmona, A., Encinas, A.M.: Solving boundary value problems on networks using equilibrium measures. J. Funct. Anal. 171, 155–176 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Benjamini, I., Schramm, O.: Harmonic functions on planar and almost planar graphs and manifolds, via circle packing. Invent. Math. 126, 565–587 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Cartwright, D.I., Woess, W.: Infinite graphs with nonconstant Dirichlet finite harmonic functions. SIAM J. Disc. Math. 73, 25–40 (1988)Google Scholar
  5. 5.
    Dodziuk, J.: Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Am. Math. Soc. 284, 787–794 (1984)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Dodziuk, J., Karp, L.: Spectral and function theory for combinatorial Laplacians. Contemp. Math. 73, 25–40 (1987)zbMATHGoogle Scholar
  7. 7.
    Dodziuk, J., Kendall, W.S.: Combinatorial Laplacians and isoperimetric inequality. Pitman Res. Notes Math. Ser. Longman Sci. Tech., Harlow 150, 68–74 (1986)Google Scholar
  8. 8.
    Holopainen, I., Soardi, P.M.: A strong Liouville theorem for p-harmonic functions on graphs. Ann. Acad. Sc. Fenn. Ser. A I Math. 22, 205–226 (1997)zbMATHGoogle Scholar
  9. 9.
    Lyons, T.: A simple criterion for transience of a reversible Markov chain. Ann. Prob. 11, 393–402 (1983)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Nash-Williams, C.St.J.A.: Random walk and electric currents in networks. Proc. Cambridge Philos. Soc. 55, 181–194 (1959)zbMATHGoogle Scholar
  11. 11.
    Pólya, G.: Über eine Aufgabe der Wahrscheinlichkeitstheorie betreffend die Irrfahrt im Strassennetz. Math. Ann. 84, 149–160 (1921)Google Scholar
  12. 12.
    Rigoli, M., Salvatori, M., Vignati, M.: Subharmonic functions on graphs. Israel J. Math. 99, 1–27 (1997)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Rigoli, M., Salvatori, M., Vignati, M.: Some remarks on the weak maximum principle. To appear on Revista Matematica IberoAmericanaGoogle Scholar
  14. 14.
    Saloff-Coste, L.: Some inequalities for superharmonic functions on graphs. Potential Anal. 6, 163–181 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Soardi, P.M.: Rough isometries and Dirichlet finite harmonic functions on graphs. Proc. Am. Math. Soc. 119, 1239–1248 (1993)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Soardi, P.M.: Potential Theory on Infinite Networks. Lecture Notes in Mathematics 1590, Springer Verlag, 1994Google Scholar
  17. 17.
    Woess, W.: Random walks on infinite graphs and groups. Cambridge Tracts in Mathematics, 138, Cambridge University Press, Cambridge, 2000Google Scholar
  18. 18.
    Yamasaki, M.: Parabolic and hyperbolic infinite networks. Hiroshima Math. J. 7, 135–146 (1977)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Dipartimento di Scienze Chimiche, Matematiche e FisicheUniversità dell’InsubriaComoItaly
  2. 2.Dipartimento di MatematicaUniversità di MilanoMilanoItaly
  3. 3.Dipartimento di MatematicaUniversità di MilanoMilanoItaly

Personalised recommendations