Abstract
We prove that certain diffusions on the Sierpinski gasket may be characterized, up to a multiplicative constant in the time scale, by a parameter α∈[0, 1/4]. The diffusions considered have the Feller property and certain natural symmetry properties, but they are not necessarily scale invariant. The case α=0 corresponds roughly speaking to one-dimensional Brownian motion and the case α=1/4 corresponds to Brownian motion on the Sierpinski gasket.
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References
Barlow, M.T. and Perkins, E.A. (1988). Brownian motion on the Sierpinski gasket. Probab. Theory Related Fields 79, 543–623.
Bochner, S. (1955). Harmonic analysis and the theory of probability, Univ. of California Press, Berkeley.
Goldstein, S. (1987). Random walks and diffusions on fractals. In: Kesten, H. (ed.) Percolation theory and ergodic theory of infinite particle systems. (IMA Math. Appl., vol. 8.) Springer, New York, p. 121–129.
Hattori, K., Hattori, T. and Watanabe, H. (1994). Asymptotically one-dimensional diffusions on the Sierpinski gasket and the abc-gaskets. Probab. Theory Related Fields 100, 85–116.
Heck, M. (1996). A perturbation result for the asymptotic behavior of matrix powers. J. Theoret. Probability 9, 647–658.
Karlin, S. (1966). A first course in stochastic processes. Academic Press, New York.
Kumagai, T. Construction and some properties of a class of non-symmetric diffusion processes on the Sierpinski gasket. In: Elworthy, K.D. and Ikeda, N. Asymptotic Problems in Probability Theory, Pitman.
Kusuoka, S. (1987). A diffusion process on a fractal. In: Ito, K. and Ikeda, N. (eds.) Symposium on Probabilistic Methods in Mathematical Physics. Proceedings Taniguchi Symposium, Katata 1985. Academic Press, Amsterdam, p. 251–274.
Lindstrøm, T. (1990). Brownian motion on nested fracrals. Mem. Amer. Math. Soc. 420.
Mode, C.J. (1971). Multi type branching processes, American Elsevier Publishing Company, New York.
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Heck, M.K. (1998). Homogeneous diffusions on the Sierpinski gasket. In: Azéma, J., Yor, M., Émery, M., Ledoux, M. (eds) Séminaire de Probabilités XXXII. Lecture Notes in Mathematics, vol 1686. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0101753
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DOI: https://doi.org/10.1007/BFb0101753
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