Gaussian Upper Bounds for Heat Kernels of Continuous Time Simple Random Walks

Matthew Folz (University of British Columbia)


We consider continuous time simple random walks with arbitrary speed measure $\theta$ on infinite weighted graphs. Write $p_t(x,y)$ for the heat kernel of this process. Given on-diagonal upper bounds for the heat kernel at two points $x_1,x_2$, we obtain a Gaussian upper bound for $p_t(x_1,x_2)$. The distance function which appears in this estimate is not in general the graph metric, but a new metric which is adapted to the random walk. Long-range non-Gaussian bounds in this new metric are also established. Applications to heat kernel bounds for various models of random walks in random environments are discussed.

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Pages: 1693-1722

Publication Date: September 12, 2011

DOI: 10.1214/EJP.v16-926


  1. S. Andres, M. T. Barlow, J.-D. Deuschel, B. M. Hambly. Invariance principle for the random conductance model. Preprint. Math. Review number not available.
  2. M. T. Barlow. Random Walks on Graphs. Unpublished manuscript. Math. Review number not available.
  3. M. T. Barlow. Random walks on supercritical percolation clusters, Ann. Probab. 32 (2004), 3024-3084. Math. Review 2006e:60146
  4. M. T. Barlow, J.-D. Deuschel. Invariance principle for the random conductance model with unbounded conductances, Ann. Probab. 38 (2010), 234-276. Math. Review 2011c:60329
  5. M. T. Barlow, R. F. Bass. Random walks on graphical Sierpinski carpets. In: Random walks and discrete potential theory, ed. M. Picardello, W. Woess, Cambridge University Press, 1999. Math. Review 2002c:60116
  6. E. A. Carlen, S. Kusuoka, D. W. Stroock. Upper bounds for symmetric Markov transition functions, Ann. I. H. Poincare-PR (1987), 245-287. Math. Review 88i:35066
  7. T. Coulhon, A Grigor'yan, F. Zucca. The discrete integral maximum principle and its applications, Tohoku Math. J. 57 (2005), 559-587. Math. Review 2007j:60070
  8. E. B. Davies. Large deviations for heat kernels on graphs, J. London Math. Soc. 47 (1993), 65-72. Math. Review 94f:58135
  9. E. B. Davies. Analysis on graphs and noncommutative geometry, J. Funct. Anal. 111 (1993), 398-430. Math. Review 93m:58110
  10. E. B. Davies. Explicit constants for Gaussian upper bounds on heat kernels, Am. J. Math. 109 (1987), 319-334. Math. Review 88g:58174
  11. T. Delmotte. Parabolic Harnack inequality and estimates of Markov chains on graphs, Rev. Mat. Iberoam. 15 (1999), 181-232. Math. Review 2000b:35103
  12. R. Frank, D. Lenz, D. Wingert. Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory. Preprint. Math. Review number not available.
  13. A. Grigor'yan. Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Differ. Geom. 45 (1997), 33-52. Math. Review 98g:58167
  14. A. Grigor'yan. Heat kernel upper bounds on a complete non-compact manifold, Rev. Math. Iberoam. 10 (1994), 395-452. Math. Review 96b:58107
  15. A. Grigor'yan, X. Huang, J. Masamune. On stochastic completeness of jump processes. To appear in Math. Z. Math. Reivew number not available.
  16. G. R. Grimmett. Percolation (2nd edition), Springer-Verlag, Berlin, 1999. Math. Review 2001a:60114
  17. W. Hebisch, L. Saloff-Coste. Gaussian estimates for Markov chains and random walks on groups, Ann. Probab. 21 (1993), 673-709. Math. Review 94m:60144
  18. W. Hebisch, L. Saloff-Coste. On the relation between elliptic and parabolic Harnack inequalities, Ann. I. Fourier 51 (2001), 1437-1481. Math. Review 2002g:58024
  19. M. Keller, D. Lenz. Unbounded Laplacians on graphs: basic spectral properties and the heat equation, Math. Model. Nat. Phenom. 5 (2010), 198-224. Math. Review 2011e:60183
  20. M. Keller, D. Lenz. Dirichlet forms and stochastic completeness of graphs and subgraphs. To appear in J. Reine. Angew. Math. Math. Review number not available.
  21. P. Mathieu, E. Remy. Isoperimetry and heat kernel decay on percolation clusters, Ann. Probab. 32 (2004), 100-128. Math. Review 2005e:60233
  22. N. Th. Varopoulos. Hardy-Littlewood theory for semigroups. J. Funct. Anal. 63 (1985), 240-260. Math. Review 87a:31011
  23. N. Varopoulos, L. Saloff-Coste, T. Coulhon. Analysis and Geometry on Groups, Cambridge University Press, Cambridge, U.K., 1992. Math. Review 95f:43008
  24. W. Woess. Random Walks on Infinite Graphs and Groups, Cambridge University Press, Cambridge, U.K., 2000. Math. Review 2001k:60006
  25. A. Weber. Analysis of the physical Laplacian and the heat flow on a locally finite graph, J. Math. Anal. Appl. 370 (2010), 146-158. Math. Review 2011f:35341

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