Skip to main content

Heat Kernel Estimates Using Effective Resistance

  • 1491 Accesses

Part of the Lecture Notes in Mathematics book series (LNMECOLE,volume 2101)

Abstract

In this chapter, we will study detailed asymptotic properties of Green functions and heat kernels using the effective resistance. Let (X, μ) be a weighted graph. We say that (X, μ) is a tree if for any l ≥ 3, there is no set of distinct points \(\{x_{i}\}_{i=1}^{l} \subset X\) such that x i x i+1 for 1 ≤ il where we set \(x_{l+1}:= x_{1}\).

Keywords

  • Green Function
  • Heat Kernel
  • Green Density
  • Weighted Graph
  • Recurrent Case

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   44.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions
Fig. 4.1

References

  1. M.T. Barlow, Diffusions on fractals, in Ecole d’été de probabilités de Saint-Flour XXV—1995. Lecture Notes in Mathematics, vol. 1690 (Springer, New York, 1998)

    Google Scholar 

  2. M.T. Barlow, Random Walks on Graphs (Cambridge University Press, to appear)

    Google Scholar 

  3. M.T. Barlow, T. Coulhon, A. Grigor’yan, Manifolds and graphs with slow heat kernel decay. Invent. Math. 144, 609–649 (2001)

    Google Scholar 

  4. M.T. Barlow, T. Coulhon, T. Kumagai, Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs. Commun. Pure Appl. Math. 58, 1642–1677 (2005)

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. B. Bollobás, Random Graphs, 2nd edn. (Cambridge University Press, Cambridge, 2001)

    CrossRef  MATH  Google Scholar 

  6. A. Grigor’yan, A. Telcs, Sub-Gaussian estimates of heat kernels on infinite graphs. Duke Math. J. 109, 452–510 (2001)

    Google Scholar 

  7. A. Grigor’yan, A. Telcs, Harnack inequalities and sub-Gaussian estimates for random walks. Math. Annalen 324, 521–556 (2002)

    Google Scholar 

  8. B.M. Hambly, T. Kumagai, Heat kernel estimates for symmetric random walks on a class of fractal graphs and stability under rough isometries, in Fractal Geometry and Applications: A Jubilee of B. Mandelbrot, Proceedings of Symposia in Pure Mathematics, vol. 72, Part 2 (American Mathematical Society, Providence, 2004), pp. 233–260

    Google Scholar 

  9. O.D. Jones, Transition probabilities for the simple random walk on the Sierpinski graph. Stoch. Process. Their Appl. 61, 45–69 (1996)

    CrossRef  MATH  Google Scholar 

  10. A. Kasue, Convergence of metric graphs and energy forms. Rev. Mat. Iberoam. 26, 367–448 (2010)

    CrossRef  MATH  MathSciNet  Google Scholar 

  11. J. Kigami, Analysis on Fractals (Cambridge University Press, Cambridge, 2001)

    CrossRef  MATH  Google Scholar 

  12. J. Kigami, Dirichlet forms and associated heat kernels on the Cantor set induced by random walks on trees. Adv. Math. 225, 2674–2730 (2010)

    CrossRef  MATH  MathSciNet  Google Scholar 

  13. J. Kigami, Resistance forms, quasisymmetric maps and heat kernel estimates. Mem. Amer. Math. Soc. 216(1015), vi+132 pp. (2012)

    Google Scholar 

  14. T. Kumagai, J. Misumi, Heat kernel estimates for strongly recurrent random walk on random media. J. Theor. Probab. 21, 910–935 (2008)

    CrossRef  MATH  MathSciNet  Google Scholar 

  15. R.S. Strichartz, Differential Equations on Fractals: A Tutorial (Princeton University Press, Princeton, 2006)

    Google Scholar 

  16. A. Telcs, The Art of Random Walks. Lecture Notes in Mathematics, vol. 1885 (Springer, Berlin, 2006)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Kumagai, T. (2014). Heat Kernel Estimates Using Effective Resistance. In: Random Walks on Disordered Media and their Scaling Limits. Lecture Notes in Mathematics(), vol 2101. Springer, Cham. https://doi.org/10.1007/978-3-319-03152-1_4

Download citation