96-152 Hattori, T., Watanabe, H.

Anisotropic random walks and the asymptotically one-dimensional diffusions on the abc-gasket (62K, LaTeX) Apr 25, 96

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Abstract. A new class of fractals, abc-gaskets, is defined and asymptotically one-dimensional diffusion processes are studied on them. The class contains the Sierpinski gasket as well as infinitely many fractals which lack non-degenerate fixed points of renormalization maps (hence are not in the class of nested fractals). The lack of non-degenerate fixed points implies that the ``standard'' diffusions are degenerate on such fractals and the standard construction of (non-degenerate) diffusion processes fails. The asymptotically one-dimensional diffusion, in contrast to the standard diffusion, is constructed on any abc-gasket by means of an unstable degenerate fixed point. To this end, the generating functions for numbers of steps of anisotropic random works on the \ags\ are analyzed, according to the line of authors' previous studies, and relevant scaling factors are calculated explicitly. In addition, a general strategy of handling random walk sequences with more than one parameters for the construction of asymptotically one-dimensional diffusion is proposed.

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