Stability of parabolic Harnack inequalities

Transactions of the American Mathematical Society

Volumn 356, Issue 4, 2004, Pages 1501-1533

  Barlow, Martin T ^a   Bass, Richard F ^b  

^a UNIVERSITY OF BRITISH COLUMBIA   (Canada)

^b UNIVERSITY OF CONNECTICUT   (United States)

Author keywords

Anomalous diffusion; Green functions; Harnack inequality; Poincar inequality; Random walks on graphs; Sobolev inequality; Volume doubling

Indexed keywords

EID: 1642375655     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0002-9947-03-03414-7     Document Type: Article

References (29)

1. [A] D.G. Aronson, Bounds on the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. 73 (1967), 890-896. MR 36:534
2. [B1] M.T. Barlow. Which values of the volume growth and escape time exponent are possible for a graph? To appear Rev. Math. Iberoamericana.
3. [BB1] M.T. Barlow, R.F. Bass, Brownian motion and harmonic analysis on Sierpinski carpets. Canad. J. Math. 51 (1999), 673-744. MR 2000i:60083
4. [BB2] M.T. Barlow, R.F. Bass. Random walks on graphical Sierpinski carpets. In: Random walks and discrete potential theory, ed. M. Piccardello, W. Woess, Symposia Mathematica XXXIX Cambridge Univ. Press, Cambridge, 1999. MR 2002c:60116
5. [BB3] M.T. Barlow, R.F. Bass. Divergence form operators on fractal-like domains. J. Funct. Analysis 175 (2000), 214-247. MR 2001i:58071
6. [BCG] M. Barlow, T. Coulhon, A. Grigor'yan. Manifolds and graphs with slow heat kernel decay. Invent. Math. 144 (2001), 609-649. MR 2002b:58029
7. [BPY] M. Barlow, J. Pitman, M. Yor. On Walsh's Brownian motions. Sém. Prob. XXIII, 275-293, Lecture Notes in Math., 1372, Springer, Berlin, 1989. MR 91a:60204
8. [BPY] M. Barlow, J. Pitman, M. Yor. On Walsh's Brownian motions. Sém. Prob. XXIII, 275-293, Lecture Notes in Math., 1372, Springer, Berlin, 1989. MR 91a:60204
9. [Da] E.B. Davies. Heat kernels and spectral theory. Cambridge University Press, 1989. MR 92a:35035
10. [D1] T. Delmotte. Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Math. Iberoamericana 15 (1999), 181-232. MR 2000b:35103
11. [D2] T. Delmotte. Graphs between the elliptic and parabolic Harnack inequalities. Potential Anal. 16 (2002), no. 2, 151-168. MR 2003b:39019
12. [FS] E.B. Fabes and D.W. Stroock, A new proof of Moser's parabolic Harnack inequality via the old ideas of Nash. Arch. Mech. Rat. Anal. 96 (1986), 327-338. MR 88b:35037
13. [FOT] M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, de Gruyter, Berlin, 1994. MR 96f:60126
14. [Gr] A.A. Grigor'yan. The heat equation on noncompact Riemannian manifolds. Math. USSR Sbornik 72 (1992), 47-77. MR 92h:58189
15. [GT1] A. Grigor'yan, A. Telcs. Sub-Gaussian estimates of heat kernels on infinite graphs. Duke Math. J. 109, (2001), 452-510. MR 2003a:35085
16. [GT2] A. Grigor'yan, A. Telcs. Harnack inequalities and sub-Gaussian estimates for random walks. Math. Annalen 324 (2002), 521-556.
17. [HK] B.M. Hambly, T. Kumagai. Heat kernel estimates for symmetric random walks on a class of fractal graphs and stability under rough isometries. To appear Proc. Symp. Pure Math.
18. [ESC] W. Hebisch, L. Saloff-Coste. On the relation between elliptic and parabolic Harnack inequalities. Ann. Inst. Fourier (Grenoble) 51 (2001), 1437-1481. MR 2002g:58024
19. [Jo] O.D. Jones. Transition probabilities for the simple random walk on the Sierpinski graph. Stock. Proc. Appl. 61 (1996), 45-69. MR 97b:60115
20. [K] S. Kusuoka: Dirichlet forms on fractals and products of random matrices. Publ. RIMS Kyoto Univ., 25, 659-680 (1989). MR 91m:60142
21. [M1] J. Moser, On Harnack's inequality for elliptic differential equations. Comm. Pure Appl. Math. 14 (1961), 577-591. MR 28:2356
22. [M2] J. Moser. On Harnack's inequality for parabolic differential equations. Comm. Pure Appl. Math. 17 (1964), 101-134. MR 28:2357
23. [M3] J. Moser, On a pointwise estimate for parabolic differential equations. Comm. Pure Appl. Math. 24 (1971), 727-740. MR 44:5603
24. [N] J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. Math. J. 80 (1958), 931-954. MR 20:6592
25. [SC] L. Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequalities. Inter. Math. Res. Notices (1992), 27-38. MR 93d:58158
26. [St] K.-T. Sturm. Analysis on local Dirichlet spaces III. The parabolic Harnack inequality. J. Math. Pures. Appl. (9) 75 (1996), 273-297. MR 97k:31010
27. [T1] A. Telcs. Local sub-Gaussian transition probability estimates, the strongly recurrent case. Electr. J. Prob. 6 (2001), no. 22, 33 pp. MR 2003a:60066
28. [T2] A. Telcs. Random walks on graphs with volume and time doubling. Preprint.
29. [W] J.B. Walsh. A diffusion with a discontinuous local time. Temps Locaux, Astérisque 52-53 (1978), 37-45.