Summary
We obtain upper and lower bounds for the transition densities of Brownian motion on nested fractals. Compared with the estimate on the Sierpinski gasket, the results require the introduction of a new exponent,d J, related to the “shortest path metric” and “chemical exponent” on nested fractals. Further, Hölder order of the resolvent densities, sample paths and local times are obtained. The results are obtained using the theory of multi-type branching processes.
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References
Athreya, K.B., Ney, P.E.: Branching processes. Berlin Heidelberg New York: Springer 1972
Bandt, C., Stanke, J.: Self-similar sets 6. Interior distance on deterministic fractals. (Preprint 1990)
Barlow, M.T.: Random walks, electrical resistance, and nested fractals. In: Elworthy, K.D., Ikeda, N. (eds.) Asymptotic problems in probability theory: stochastic models and diffusions on fractals, pp. 131–157. Montreal: Pitman 1993
Barlow, M.T., Bass, R.F.: Transition densities for Brownian motion on the Sierpinski carpet. Probab. Theory Relat. Fields91, 307–330 (1992)
Barlow, M.T., Perkins, E.A.: Brownian motion on the Sierpinski gasket. Probab. Theory Relat. Fields79, 543–623 (1988)
Burdzy, K.: Percolation dimension of fractals. J. Math. Anal. Appl.145, 282–288 (1990)
Fitzsimmons, P.J., Hambly, B.M., Kumagai, T.: Transition density estimates for Brownian motion on weighted nested fractals. (Preprint 1993)
Fukushima, M.: Dirichlet forms, diffusion processes and spectral dimensions for nested fractals. In: Albeverio, Fenstad, Holden and Lindstrøm (eds.) Ideas and methods in mathematical analysis, stochastics, and applications, in memory of R. Høegh-Krohn, vol. 1. pp. 151–161. Cambridge: Cambridge University Press 1992
Havlin, S., Ben-Avraham, D.: Diffusion in disordered media. Adv. Phys.36, 695–798 (1987)
Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J.30 (1981)
Kumagai, T.: Construction and some properties of a class of non-symmetric diffusion processes on the Sierpinski gasket. In: Elworthy, K.D., Ikeda, N. (eds.) Asymptotic problems in probability theory: stochastic models and diffusions on fractals, pp. 219–247. Montreal: Pitman 1993
Kusuoka, S.: Diffusion processes on nested fractals. (Lect. Notes Math.) Berlin Heidelberg New York: Springer (to appear)
Lindstrøm, T.: Brownian motion on nested fractals. Mem. Am. Math. Soc.420 (1990)
Shima, T.: Lifshitz tails for random Schrödinger operators on nested fractals. Osaka J. Math.29-4, 749–770 (1992)
Yokoi, H.: Denumerable Markov chains and nested fractals. Master thesis at Tohoku University (1991)
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Kumagai, T. Estimates of transition densities for Brownian motion on nested fractals. Probab. Th. Rel. Fields 96, 205–224 (1993). https://doi.org/10.1007/BF01192133
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DOI: https://doi.org/10.1007/BF01192133