Divergence Form Operators on Fractal-like Domains

Journal of Functional Analysis

Volumn 175, Issue 1, 2000, Pages 214-247

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

^a UNIVERSITY OF BRITISH COLUMBIA   (Canada)

^b UNIVERSITY OF CONNECTICUT   (United States)

Author keywords

Harnack inequality; divergence form; Sierpinski carpet; fractals; heat equation; elliptic operators; fundamental solution; diffusions

Indexed keywords

EID: 0040822200     PISSN: 00221236     EISSN: None     Source Type: Journal    
DOI: 10.1006/jfan.2000.3597     Document Type: Article

References (24)

1. Aronson D. G. Bounds on the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. 73:1967;890-896.
2. Auscher P., Coulhon T. Gaussian lower bounds for random walks from elliptic regularity. Ann. Inst. H. Poincaré Probab. Statist. 35:1999;605-630.
3. Barlow M. T., Bass R. F. The construction of Brownian motion on the Sierpinski carpet. Ann. Inst. H. Poincaré Probab. Statist. 25:1989;225-257.
4. Barlow M. T., Bass R. F. Transition densities for Brownian motion on the Sierpinski carpet. Probab. Theory Related Fields. 91:1992;307-330.
5. Barlow M. T., Bass R. F. Brownian motion and harmonic analysis on Sierpinski carpets. Canad. J. Math. 51:1999;673-744.
6. Barlow M. T., Bass R. F. Random walks on graphical Sierpinski carpets. Piccardello M., Woess W. Random Walks and Discrete Potential Theory. Symposia Mathematica. XXXIX:1999;Cambridge Univ. Press, Cambridge.
7. M. Barlow, T. Coulhon, and, A. Grigor'yan, Manifolds and graphs with slow heat kernel decay, preprint.
8. Delmotte T. Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoamer. 15:1999;181-232.
9. Fabes E. B., Stroock D. W. A new proof of Moser's parabolic Harnack inequality via the old ideas of Nash. Arch. Rational. Mech. Anal. 96:1986;327-338.
10. Fukushima M., Oshima Y., Takeda M. Dirichlet Forms and Symmetric Markov Processes. 1994;de Gruyter, Berlin.
11. Grigor'yan A. A. The heat equation on noncompact Riemannian manifolds. Math. USSR Sb. 72:1992;47-77.
12. Hutchinson J. E. Fractals and self-similarity. Indiana Univ. Math. J. 30:1981;713-747.
13. Jerison D. The Poincaré inequality for vector fields satisfying Hörmander's condition. Duke Math. J. 53:1986;503-523.
14. Jones O. D. Transition probabilities for the simple random walk on the Sierpinski graph. Stochastic Process. Appl. 61:1996;45-69.
15. Kusuoka S. Dirichlet forms on fractals and products of random matrices. Publ. Res. Inst. Math. Sci. 25:1989;659-680.
16. Mandelbrot B. The Fractal Geometry of Nature. 1982;Freeman, San Francisco.
17. McGilllivray I. Some Applications of Dirichlet Forms in Probability Theory. 1992;Cambridge Univ. Press, Cambridge.
18. Moser J. On Harnack's inequality for elliptic differential equations. Comm. Pure Appl. Math. 14:1961;577-591.
19. Moser J. On a pointwise estimate for parabolic differential equations. Comm. Pure Appl. Math. 24:1971;727-740.
20. Osada H. Isoperimetric dimension and estimates of heat kernels of pre-Sierpinski carpets. Probab. Theory Related Fields. 86:1990;469-490.
21. Pardoux E., Williams R. J. Symmetric reflected diffusions. Ann. Inst. H. Poincaré Anal. Non Linéaire. 30:1994;13-62.
22. Saloff-Coste L. A note on Poincaré, Sobolev, and Harnack inequalities. Internat. Math. Res. Notices. 1992;27-38.
23. Sierpinski W. Sur une courbe cantorienne qui contient une image biunivoque et continue de toute courbe donnée. C.R. Acad. Sci. paris. 162:1916;629-632.
24. Sturm K.-T. Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality. J. Math. Pures Appl. (9). 75:1996;273-297.