Rotation Invariance and Characterization of a Class of Self-Similar Diffusion Processes on the Sierpinski Gasket

In [B.P], Barlow-Perkins succeeded in the characterization of the Brownian motion on the Sierpinski gasket. They proved that the diffusion on the gasket which has local translation and reflection invariance is a constant time change of the Brownian motion. On the other hand, Kumagai [Kum] introduced a class of Feller diffusions which is invariant under the operation of local rotation. These diffusions are called p-stream diffusions on the Sierpinski gasket, which contains Brownian motion as a typical case. They were constructed as a limit of a sequence of random walks which has some consistency (called decimation property). In this paper, we will characterize these Feller diffusions. In fact, the non-degenerate self similar Feller diffusion which has local rotation invariance is a constant time change of some p-stream diffusion. In general, the problem of this type is essentially reduced to show the uniqueness of the fixed point for some non-linear map. In Section 1, we briefly introduce the p-stream diffusions and give some properties of them. In Section 2, we characterize these diffusions. In Section 3, we give some remarks for the existence of non-symmetric Feller diffusions on some fractals.