Anisotropic random walks and asymptotically one-dimensional diffusion on abc-gaskets

Asymptotically one-dimensional diffusion processes are studied on the class of fractals called abc-gaskets. The class is a set of certain variants of the Sierpiński gasket containing infinitely many fractals without any nondegenerate fixed point of renormalization maps. While the "standard" method of constructing diffusions on the Sierpiński gasket and on nested fractals relies on the existence of a nondegenerate fixed point and hence it is not applicable to all abc-gaskets, the asymptotically one-dimensional 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 walks on the abc-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.