Anisotropic random walks and asymptotically one-dimensional diffusion onabc-gaskets

Abstract

Asymptotically one-dimensional diffusion processes are studied on the class of fractals calledabc-gaskets. The class is a set of certain variants of the Sierpinski gasket containing infinitely many fractals without any nondegenerate fixed point of renoramalization maps. While the “standard” method of constructing diffusions on the Sierpinski gasket and on nested fractals relies on the existence of a nondegenerate fixed point and hence it is not applicable to allabc-gaskets, the asymptotically one-dimensional diffusion is constructed on anyabc-gasket by means of an unstable degenerate fixed point. To this end, the generating functions for numbers of steps of anisotropic random walks on theabc-gaskets are analyzed, along the line of the authors' previous studies. In addition, a general stategy of handling random walk sequences with more than one parameter for the construction of asymptotically one-dimensional diffusion is proposed.