Random walk on graphs with regular resistance and volume growth

 Access to full text

 Full (PDF)

1. D. G. Aronson. Non-negative solutions of linear parabolic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci (3) 22 (1968) 607–694; Addendum 25 (1971) 221–228. Zbl0223.35046MR435594
2. M. T. Barlow. Diffusion on fractals. Lectures on Probability Theory and Statistics, Ecole d’été de probabilités de Saint-Flour XXV -1995, pp. 1–121. Lecture Notes in Math. 1690. Springer, Berlin, 1998. Zbl0916.60069MR1668115
3. M. T. Barlow. Some remarks on the elliptic Harnack inequality. Bull. Lond. Math. Soc. 37 (2005) 200–208. Zbl1067.31002MR2119019
4. M. T. Barlow and R. Bass. Stability of the parabolic Harnack inequalities. Trans. Amer. Math. Soc. 356 (2003) 1501–1533. Zbl1034.60070MR2034316
5. M. T. Barlow, R. Bass and T. Kumagai. Stability of parabolic Harnack inequalities on metric measure spaces. J. Math. Soc. Japan. 58 (2006) 485–519. Zbl1102.60064MR2228569
6. M. Barlow, T. Coulhon and A. Grigor’yan. Manifolds and graphs with slow heat kernel decay. Invent. Math. 144 (2001) 609–649. Zbl1003.58025MR1833895
7. M. T. Barlow, T. Coulhon and T. Kumagai. Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs. Comm. Pure Appl. Math. 58 (2005) 1642–1677. Zbl1083.60060MR2177164
8. T. Coulhon and A. Grigor’yan. Random walks on graphs with regular volume growth. Geom. Funct. Anal. 8 (1998) 656–701. Zbl0918.60053MR1633979
9. E. B. Davies. Heat Kernels and Spectral Theory. Cambridge University Press, 1989. Zbl0699.35006MR990239
10. T. Delmotte. Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoamericana 1 (1999) 181–232. Zbl0922.60060MR1681641
11. T. Delmotte. Graphs between the elliptic and parabolic Harnack inequalities. Potential Anal. 16 (2002) 151–168. Zbl1081.39012MR1881595
12. E. B. Fabes and D. W. Stroock. A new proof of Moser’s parabolic Harnack inequality using an old idea of Nash. Arch. Rational Mech. Anal. 96 (1986) 327–338. Zbl0652.35052MR855753
13. A. Grigor’yan. Heat equation on a non-compact Riemannian manifold. Math. USSR Sb. 72 (1992) 47–77. Zbl0776.58035
14. A. Grigor’yan. Heat kernel upper bounds on fractal spaces. Unpublished manuscript. 
15. A. Grigor’yan and A. Telcs. Sub-Gaussian estimates of heat kernels on infinite graphs. Duke Math. J. 109 (2001) 452–510. Zbl1010.35016MR1853353
16. A. Grigor’yan and A. Telcs. Harnack inequalities and sub-Gaussian estimates for random walks. Math. Ann. 324 (2002) 521–556. Zbl1011.60021MR1938457
17. A. Grigor’yan and A. Telcs. Heat kernel estimates in measure metric spaces. Unpublished manuscript. 
18. M. Gromov. Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. 53 (1981) 57–73. Zbl0474.20018MR623534
19. B. Hambly and T. Kumagai. Heat kernel estimates for symmetric random walks on a class of fractal graphs and stability under rough isometries. In Proc. of Symposia in Pure Math. 72. Part 2, pp. 233–260. Amer. Math. Soc., Providence, RI, 2004. Zbl1065.60041MR2112125
20. M. Hino and J. A. Ramírez. Small-time Gaussian behavior of symmetric diffusion semigroups. Ann. Probab. 31 (2003) 1254–1295. Zbl1085.31008MR1988472
21. W. Hebisch and L. Saloff-Coste. On the relation between elliptic and parabolic Harnack inequalities. Ann. Inst. Fourier 51 (2001) 1437–1481. Zbl0988.58007MR1860672
22. O. D. Jones. Transition probabilities for the simple random walk on the Sierpinski graph. Stochastic Process Appl. 61 (1996) 45–69. Zbl0853.60058MR1378848
23. T. Kumagai and K.-T. Sturm. Construction of diffusion processes on fractals, d-sets, and general metric measure spaces, J. Math. Kyoto Univ. 45 (2005) 307–327. Zbl1086.60052MR2161694
24. P. Li and J. Wang. Mean value inequalities. Indiana Univ. Math. J. 48 (1999) 1257–1283. Zbl1003.58026MR1757075
25. P. Li and S.-T. Yau. On the parabolic kernel of the Schrödinger operator. Acta Math. 156 (1986) 153–201. Zbl0611.58045MR834612
26. J. Moser. On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. 16 (1964) 101–134. Zbl0149.06902MR159139
27. J. Moser. On Harnack’s theorem for parabolic differential equations. Comm. Pure Appl. Math. 24 (1971) 727–740. Zbl0227.35016MR288405
28. J. R. Norris. Heat kernel asymptotics and the distance function in Lipschitz Riemannian manifolds. Acta Math. 179 (1997) 79–103. Zbl0912.58041MR1484769
29. C. Sabot. Existence and uniqueness of diffusions on finitely ramified self-similar fractals. Ann. Sci. École Norm. Sup (4) 30 (1997) 605–673. Zbl0924.60064MR1474807
30. L. Saloff-Coste. A note on Poincaré, Sobolev and Harnack inequalities. Internat. Math. Res. 2 (1992) 27–38. Zbl0769.58054MR1150597
31. C.-J. Sung. Parabolic super mean value inequality. Proc. Amer. Math. Soc. 130 (2002) 3401–3408. Zbl1010.58021MR1913020
32. R. S. Strichartz. Fractafolds based on the Sierpinski gasket and their spectra. Trans. Amer. Math. Soc. 355 (2003) 4019–4043. Zbl1041.28006MR1990573
33. K.-T. Sturm. Analysis on local Dirichlet spaces III. The parabolic Harnack inequality. J. Math. Pures Appl. (9) 75 (1996) 273–297. Zbl0854.35016MR1387522
34. A. Telcs. Volume and time doubling of graphs and random walk, the strongly recurrent case. Comm. Pure Appl. Math. (2001) 975–1018. Zbl1021.60037MR1829530
35. A. Telcs. Some notes on the Einstein relation. J. Stat. Phys. 122 (2006) 617–645. Zbl1149.82025MR2213945
36. A. Telcs. Random walks on graphs with volume and time doubling. Revista Mat. Iberoamericana 22 (2006) 17–54. Zbl1118.60062MR2267312
37. A. Telcs. The Art of Random Walks. Springer, Berlin, 2006. Zbl1104.60003MR2240535
38. S. R. S. Varadhan. On the behavior of the fundamental solution of the heat equation with variable coefficients. Comm. Pure Appl. Math. 20 (1967) 431–455. Zbl0155.16503MR208191