Stability of parabolic Harnack inequalities

Authors:
Martin T. Barlow and Richard F. Bass

Journal:
Trans. Amer. Math. Soc. **356** (2004), 1501-1533

MSC (2000):
Primary 60J27; Secondary 60J35, 31B05

Published electronically:
September 22, 2003

MathSciNet review:
2034316

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a graph with weights for which a parabolic Harnack inequality holds with space-time 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.

**[A]**D. G. Aronson,*Bounds for the fundamental solution of a parabolic equation*, Bull. Amer. Math. Soc.**73**(1967), 890–896. MR**0217444**, 10.1090/S0002-9904-1967-11830-5**[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]**Martin T. Barlow and Richard F. Bass,*Brownian motion and harmonic analysis on Sierpinski carpets*, Canad. J. Math.**51**(1999), no. 4, 673–744. MR**1701339**, 10.4153/CJM-1999-031-4**[BB2]**Martin T. Barlow and Richard F. Bass,*Random walks on graphical Sierpinski carpets*, Random walks and discrete potential theory (Cortona, 1997) Sympos. Math., XXXIX, Cambridge Univ. Press, Cambridge, 1999, pp. 26–55. MR**1802425****[BB3]**Martin T. Barlow and Richard F. Bass,*Divergence form operators on fractal-like domains*, J. Funct. Anal.**175**(2000), no. 1, 214–247. MR**1774857**, 10.1006/jfan.2000.3597**[BCG]**Martin Barlow, Thierry Coulhon, and Alexander Grigor’yan,*Manifolds and graphs with slow heat kernel decay*, Invent. Math.**144**(2001), no. 3, 609–649. MR**1833895**, 10.1007/s002220100139**[BPY]**Martin Barlow, Jim Pitman, and Marc Yor,*On Walsh’s Brownian motions*, Séminaire de Probabilités, XXIII, Lecture Notes in Math., vol. 1372, Springer, Berlin, 1989, pp. 275–293. MR**1022917**, 10.1007/BFb0083979**[Da]**E. B. Davies,*Heat kernels and spectral theory*, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1990. MR**1103113****[D1]**Thierry Delmotte,*Parabolic Harnack inequality and estimates of Markov chains on graphs*, Rev. Mat. Iberoamericana**15**(1999), no. 1, 181–232. MR**1681641**, 10.4171/RMI/254**[D2]**Thierry Delmotte,*Graphs between the elliptic and parabolic Harnack inequalities*, Potential Anal.**16**(2002), no. 2, 151–168. MR**1881595**, 10.1023/A:1012632229879**[FS]**E. B. Fabes and D. W. Stroock,*A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash*, Arch. Rational Mech. Anal.**96**(1986), no. 4, 327–338. MR**855753**, 10.1007/BF00251802**[FOT]**Masatoshi Fukushima, Yōichi Ōshima, and Masayoshi Takeda,*Dirichlet forms and symmetric Markov processes*, de Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 1994. MR**1303354****[Gr]**A. A. Grigor′yan,*The heat equation on noncompact Riemannian manifolds*, Mat. Sb.**182**(1991), no. 1, 55–87 (Russian); English transl., Math. USSR-Sb.**72**(1992), no. 1, 47–77. MR**1098839****[GT1]**Alexander Grigor′yan and Andras Telcs,*Sub-Gaussian estimates of heat kernels on infinite graphs*, Duke Math. J.**109**(2001), no. 3, 451–510. MR**1853353**, 10.1215/S0012-7094-01-10932-0**[GT2]**A. Grigor'yan, A. Telcs. Harnack inequalities and sub-Gaussian estimates for random walks.*Math. Annalen***324**(2002), 521-556.**[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 and L. Saloff-Coste,*On the relation between elliptic and parabolic Harnack inequalities*, Ann. Inst. Fourier (Grenoble)**51**(2001), no. 5, 1437–1481 (English, with English and French summaries). MR**1860672****[Jo]**Owen Dafydd Jones,*Transition probabilities for the simple random walk on the Sierpiński graph*, Stochastic Process. Appl.**61**(1996), no. 1, 45–69. MR**1378848**, 10.1016/0304-4149(95)00074-7**[K]**Shigeo Kusuoka,*Dirichlet forms on fractals and products of random matrices*, Publ. Res. Inst. Math. Sci.**25**(1989), no. 4, 659–680. MR**1025071**, 10.2977/prims/1195173187**[M1]**Jürgen Moser,*On Harnack’s theorem for elliptic differential equations*, Comm. Pure Appl. Math.**14**(1961), 577–591. MR**0159138****[M2]**Jürgen Moser,*A Harnack inequality for parabolic differential equations*, Comm. Pure Appl. Math.**17**(1964), 101–134. MR**0159139****[M3]**J. Moser,*On a pointwise estimate for parabolic differential equations*, Comm. Pure Appl. Math.**24**(1971), 727–740. MR**0288405****[N]**J. Nash,*Continuity of solutions of parabolic and elliptic equations*, Amer. J. Math.**80**(1958), 931–954. MR**0100158****[SC]**L. Saloff-Coste,*A note on Poincaré, Sobolev, and Harnack inequalities*, Internat. Math. Res. Notices**2**(1992), 27–38. MR**1150597**, 10.1155/S1073792892000047**[St]**K. T. Sturm,*Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality*, J. Math. Pures Appl. (9)**75**(1996), no. 3, 273–297. MR**1387522****[T1]**András Telcs,*Local sub-Gaussian estimates on graphs: the strongly recurrent case*, Electron. J. Probab.**6**(2001), no. 22, 33 pp. (electronic). MR**1873299**, 10.1214/EJP.v6-95**[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***52-53**(1978), 37-45.

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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/S0002-9947-03-03414-7

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