Tohoku Mathematical Journal

The discrete integral maximum principle and its applications

Thierry Coulhon, Alexander Grigor'yan, and Fabio Zucca

Full-text: Open access

Abstract

We prove an integral maximum principle for random walks on graphs, and give several applications to pointwise estimates of their transition probabilities, including the time-dependent case.

Article information

Source
Tohoku Math. J. (2), Volume 57, Number 4 (2005), 559-587.

Dates
First available in Project Euclid: 23 February 2006

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1140727073

Digital Object Identifier
doi:10.2748/tmj/1140727073

Mathematical Reviews number (MathSciNet)
MR2203547

Zentralblatt MATH identifier
1096.60023

Subjects
Primary: 60J15
Secondary: 35B50: Maximum principles 39A12: Discrete version of topics in analysis

Keywords
Discrete integral maximum principle random walks on graphs upper and lower estimates for a discrete heat kernel time-dependent random walks random walks on percolation clusters

Citation

Coulhon, Thierry; Grigor'yan, Alexander; Zucca, Fabio. The discrete integral maximum principle and its applications. Tohoku Math. J. (2) 57 (2005), no. 4, 559--587. doi:10.2748/tmj/1140727073. https://projecteuclid.org/euclid.tmj/1140727073


Export citation

References

  • D. G. Aronson, Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc. 73 (1967), 890--896.
  • P. Auscher and T. Coulhon, Gaussian lower bounds for random walks from elliptic regularity, Ann. Inst. H. Poincaré Probab. Statist. 35 (1999), 605--630.
  • P. Auscher, T. Coulhon and A. Grigor'yan, ed., ``Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002)'', Contemp. Math. 338. Amer. Math Soc., Providence, R.I., 2003.
  • M. T. Barlow, Random walks on supercritical percolation clusters, Ann. Probab. 32 (2004), 3024--3084.
  • T.K. Carne, A transmutation formula for Markov chains, Bull. Sci. Math. (2) 109 (1985), 399--405.
  • T. Coulhon, Analysis on infinite graphs with regular volume growth, Random walks and discrete potential theory (Cortona, 1997), 165--187, Sympos. Math. XXXIX, Cambridge Univ. Press, Cambridge, 1999.
  • T. Coulhon, Random walks and geometry on infinite graphs, Lecture notes on analysis in metric spaces (Trento, 1999), 5--30, Appunti Corsi Tenuti Docenti Sc., Scuola Norm. Sup., Pisa, 2000.
  • T. Coulhon and A. Grigor'yan, On-diagonal lower bounds for heat kernels and Markov chains, Duke Math. J. 89 (1997), 133--199.
  • T. Coulhon and A. Grigor'yan, Random walks on graphs with regular volume growth, Geom. Funct. Anal. 8 (1998), 656--701.
  • T. Coulhon and A. Grigor'yan, Pointwise estimates for transition probabilities of random walks on infinite graphs, Fractals in Graz 2001, 119--134, Trends Math., Birkäuser, Basel, 2003.
  • T. Coulhon and L. Saloff-Coste, Puissances d'un opérateur régularisant, Ann. Inst. H. Poincaré Probab. Statist. 26 (1990), 419--436.
  • E. B. Davies, Heat kernel bounds, conservation of probability and the Feller property, J. Anal. Math. 58 (1992), 99--119.
  • T. Delmotte, Parabolic Harnack inequality and estimates of Markov chains on graphs, Rev. Mat. Iberoamericana 15 (1999), 181--232.
  • A. Grigor'yan, Heat kernel upper bounds on a complete non-compact manifold, Rev. Mat. Iberoamericana 10 (1994), 395--452.
  • A. Grigor'yan, Integral maximum principle and its applications, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), 353--362.
  • A. Grigor'yan, Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Differential Geom. 45 (1997), 33--52.
  • A. Grigor'yan, Estimates of heat kernels on Riemannian manifolds, Spectral theory and geometry, (Edinburgh 1998), 140--225, London Math. Soc. Lecture Note Ser. 273 Cambridge Univ. Press, Cambridge, 1999.
  • W. Hebisch and L. Saloff-Coste, Gaussian estimates for Markov chains and random walks on groups, Ann. Probab. 21 (1993), 673--709.
  • P. Li and S.-T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), 153--201.
  • P. Mathieu and E. Remy, Décroissance du noyau de la chaleur et isopérimétrie sur un amas de percolation, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), 927--931.
  • P. Mathieu and E. Remy, Isoperimetry and heat kernel decay on percolation clusters, Ann. Probab. 32 (2004), 100--128.
  • Ch. Pittet and L. Saloff-Coste, A survey on the relationship between volume growth, isoperimetry, and the behavior of simple random walk on Cayley graphs, with examples, unpublished manuscript.
  • D. W. Stroock, Estimates on the heat kernel for the second order divergence form operators, Probability theory (Singapore, 1989), 29--44, de Gruyter, Berlin, 1992.
  • N. Th. Varopoulos, Long range estimates for Markov chains, Bull. Sci. Math. (2) 109 (1985), 225--252.
  • N. Th. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and geometry on groups, Cambridge Tracts in Math. 100, Cambridge Univ. Press, Cambridge, 1992.
  • W. Woess, Random walks on infinite graphs and groups---a survey on selected topics, Bull. London Math. Soc. 26 (1994), 1--60.