Summary
Upper and lower bounds are obtained for the transition densitiesp(t, x, y) of Brownian motion on the Sierpinski carpet. These are of the same form as those which hold for the Sierpinski gasket. In addition, the joint continuity ofp(t, x, y) is proved, the existence of the spectral dimension is established, and the Einstein relation, connecting the spectral dimension, the Hausdorff dimension and the resistance exponent, is shown to hold.
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References
[BS] Bandt, C., Stahnke, J.: Self-similar sets 6. Interior distance on deterministic fractals. (Preprint 1990)
[B] Barlow, M.T.: Necessary and sufficient conditions for the continuity of local time of Lévy processes. Ann. Probab.16, 1389–1427 (1988)
[BB1] Barlow, M.T., Bass, R.F.: The construction of Brownian motion on the Sierpinski carpet. Ann. Inst. Henri Poincaré25, 225–257 (1989)
[BB2] Barlow, M.T., Bass, R.F.: Local times for Brownian motion on the Sierpinski carpet. Probab. Theory Relat. Fields85, 91–104 (1990)
[BB3] Barlow, M.T., Bass, R.F.: On the resistance of the Sierpinski carpet. Proc. R. Soc. Lond Ser. A431, 345–360 (1990)
[BBS] Barlow, M.T., Bass, R.F., Sherwood, J.D.: Resistance and spectral dimension of Sierpinski carpets. J. Phys. A.23, L253-L258 (1990)
[BP] Barlow, M.T., Perkins, E.A.: Brownian motion on the Sierpinski gasket. Probab. Theory Relat. Fields79, 543–623 (1988)
[BH] Bass, R.F., Hsu, P.: Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains. Ann. Probab. (to appear)
[FaS] Fabes, E.B., Stroock, D.W.: A new proof of Moser's inequality using the old ideas of Nash. Arch. Ration. Mech. Anal.96, 328��338 (1986)
[F] Fukushima, M.: Dirichlet forms and Markov processes. Tokyo: Kodansha, 1980
[FS] Fukushima, M., Shima, T.: On a spectral analysis for the Sierpinski gasket. J. Potential Anal. (to appear)
[HBA] Havlin, S., Ben-Avraham, D.: Diffusion in disordered media. Adv. Phys.36, 695–798 (1987)
[H] Hambly, B.M.: On the limiting distribution of a supercritical branching process in a random environment. (Preprint 1990).
[KR] Krein, M.G., Rutman, M.A.: Linear operators leaving invariant a cone in a Banach space. Am. Math. Soc. Sel. Transl. Ser. 1.10, 199–325 (1962)
[L] Lindstrøm, T.: Brownian motion on nested fractals. Mem. Am. Math. Soc.240 (1990)
[M] Marcus, M.B., Rosen, J.: Sample path properties of the local times of strongly symmetric Markov processes via Gaussian processes. (Preprint 1990)
[O] Osada, H.: Isoperimetric dimension and estimates of heat kernels of pre-Sierpinski carpets. Probab. Theory Relat. Fields86, 469–490 (1990)
[RT] Rammal, R., Toulouse, G.: Random walks on fractal structures and percolation clusters. J. Phys. Lett.44, L13-L22 (1983)
[W] Watanabe, H.: Spectral dimension of a wire network. J. Phys. A.18, 2807–2823 (1985)
[Y] Yosida, K.: Lectures on differential and integral equations. New York: Interscience 1960
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Research partially supported by NSF Grant DMS 88-22053
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Barlow, M.T., Bass, R.F. Transition densities for Brownian motion on the Sierpinski carpet. Probab. Th. Rel. Fields 91, 307–330 (1992). https://doi.org/10.1007/BF01192060
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DOI: https://doi.org/10.1007/BF01192060