Stability of parabolic Harnack inequalities
Authors:
Martin T. Barlow and Richard F. Bass
Journal:
Trans. Amer. Math. Soc. 356 (2004), 15011533
MSC (2000):
Primary 60J27; Secondary 60J35, 31B05
Published electronically:
September 22, 2003
MathSciNet review:
2034316
Fulltext PDF Free Access
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Abstract: Let be a graph with weights for which a parabolic Harnack inequality holds with spacetime scaling exponent . Suppose is another set of weights that are comparable to . We prove that this parabolic Harnack inequality also holds for with the weights . We also give stable necessary and sufficient conditions for this parabolic Harnack inequality to hold.
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 [B1]
 M.T. Barlow. Which values of the volume growth and escape time exponent are possible for a graph? To appear Rev. Math. Iberoamericana.
 [BB1]
 M.T. Barlow, R.F. Bass, Brownian motion and harmonic analysis on Sierpinski carpets. Canad. J. Math. 51 (1999), 673744. MR 2000i:60083
 [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
 [BB3]
 M.T. Barlow, R.F. Bass. Divergence form operators on fractallike domains. J. Funct. Analysis 175 (2000), 214247. MR 2001i:58071
 [BCG]
 M. Barlow, T. Coulhon, A. Grigor'yan. Manifolds and graphs with slow heat kernel decay. Invent. Math. 144 (2001), 609649. MR 2002b:58029
 [BPY]
 M. Barlow, J. Pitman, M. Yor. On Walsh's Brownian motions. Sém. Prob. XXIII, 275293, Lecture Notes in Math., 1372, Springer, Berlin, 1989. MR 91a:60204
 [Da]
 E.B. Davies. Heat kernels and spectral theory. Cambridge University Press, 1989. MR 92a:35035
 [D1]
 T. Delmotte. Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Math. Iberoamericana 15 (1999), 181232. MR 2000b:35103
 [D2]
 T. Delmotte. Graphs between the elliptic and parabolic Harnack inequalities. Potential Anal. 16 (2002), no. 2, 151168. MR 2003b:39019
 [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), 327338. MR 88b:35037
 [FOT]
 M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet Forms and Symmetric Markov Processes. de Gruyter, Berlin, 1994. MR 96f:60126
 [Gr]
 A.A. Grigor'yan. The heat equation on noncompact Riemannian manifolds. Math. USSR Sbornik 72 (1992), 4777. MR 92h:58189
 [GT1]
 A. Grigor'yan, A. Telcs. SubGaussian estimates of heat kernels on infinite graphs. Duke Math. J. 109, (2001), 452510. MR 2003a:35085
 [GT2]
 A. Grigor'yan, A. Telcs. Harnack inequalities and subGaussian estimates for random walks. Math. Annalen 324 (2002), 521556.
 [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.
 [HSC]
 W. Hebisch, L. SaloffCoste. On the relation between elliptic and parabolic Harnack inequalities. Ann. Inst. Fourier (Grenoble) 51 (2001), 14371481. MR 2002g:58024
 [Jo]
 O.D. Jones. Transition probabilities for the simple random walk on the Sierpinski graph. Stoch. Proc. Appl. 61 (1996), 4569. MR 97b:60115
 [K]
 S. Kusuoka: Dirichlet forms on fractals and products of random matrices. Publ. RIMS Kyoto Univ., 25, 659680 (1989). MR 91m:60142
 [M1]
 J. Moser, On Harnack's inequality for elliptic differential equations. Comm. Pure Appl. Math. 14 (1961), 577591. MR 28:2356
 [M2]
 J. Moser. On Harnack's inequality for parabolic differential equations. Comm. Pure Appl. Math. 17 (1964), 101134. MR 28:2357
 [M3]
 J. Moser, On a pointwise estimate for parabolic differential equations. Comm. Pure Appl. Math. 24 (1971), 727740. MR 44:5603
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 J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. Math. J. 80 (1958), 931954. MR 20:6592
 [SC]
 L. SaloffCoste, A note on Poincaré, Sobolev, and Harnack inequalities. Inter. Math. Res. Notices (1992), 2738. MR 93d:58158
 [St]
 K.T. Sturm. Analysis on local Dirichlet spaces III. The parabolic Harnack inequality. J. Math. Pures. Appl. (9) 75 (1996), 273297. MR 97k:31010
 [T1]
 A. Telcs. Local subGaussian transition probability estimates, the strongly recurrent case. Electr. J. Prob. 6 (2001), no. 22, 33 pp. MR 2003a:60066
 [T2]
 A. Telcs. Random walks on graphs with volume and time doubling. Preprint.
 [W]
 J.B. Walsh. A diffusion with a discontinuous local time. Temps Locaux, Astérisque 5253 (1978), 3745.
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Additional Information
Martin T. Barlow
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, Canada V6T 1Z2
Email:
barlow@math.ubc.ca
Richard F. Bass
Affiliation:
Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
Email:
bass@math.uconn.edu
DOI:
http://dx.doi.org/10.1090/S0002994703034147
PII:
S 00029947(03)034147
Keywords:
Harnack inequality,
random walks on graphs,
volume doubling,
Green functions,
Poincar\'{e} inequality,
Sobolev inequality,
anomalous diffusion
Received by editor(s):
January 24, 2003
Published electronically:
September 22, 2003
Additional Notes:
The first author’s research was partially supported by an NSERC (Canada) grant, and by CNRS (France)
The second author’s research was partially supported by NSF Grant DMS 9988486
Article copyright:
© Copyright 2003
American Mathematical Society
