Degenerate limits for one-parameter families of non-fixed-point diffusions on fractals

Abstract

The Sierpinski gasket is known to support an exotic stochastic process called the asymptotically one-dimensional diffusion. This process displays local anisotropy, as there is a preferred direction of motion which dominates at the microscale, but on the macroscale we see global isotropy in that the process will behave like the canonical Brownian motion on the fractal. In this paper we analyse the microscale behaviour of such processes, which we call non-fixed point diffusions, for a class of fractals and show that there is a natural limit diffusion associated with the small scale asymptotics. This limit process no longer lives on the original fractal but is supported by another fractal, which is the Gromov-Hausdorff limit of the original set after a shorting operation is performed on the dominant microscale direction of motion. We establish the weak convergence of the rescaled diffusions in a general setting and use this to answer a question raised in [15] about the ultraviolet limit of the asymptotically one-dimensional diffusion process on the Sierpinski gasket.

Dirichlet forms and critical exponents on fractals

“…The construction of the energy form by the reverse recursive method was implicitly used by Hattori, Hattori and Watanabe on the Sierpinski gasket [15] through a probability consideration, and they call the limit an asymptotically one dimensional diffusion. This diffusion was investigated further by Hambly and Kumagai [12] on some other nested fractals (see also [10,14]). Recently, in [8], the authors gave a detail study of this method on the Sierpinski gasket from an analytic point of view; they showed a dichotomy result that for any initial data, the Dirichlet forms obtained are either the standard or the one in [15].…”

“…The other construction, we call it reverse recursive method, is to fix an initial data at V 0 , and iterate this to V n to obtain a sequence of compatible networks. This method first appeared in a probabilistic study by Hattori, Hattori and Watanabe [15] on the Sierpinski gasket K (abc-gasket), they showed that there is an asymptotically one-dimensional diffusion process on K. Some further development and extensions can be found in [10,11,12,14] by Hambly et al, and in [8] by the authors.…”

On a recursive construction of Dirichlet form on the Sierpiński gasket

“…We remark that in another investigation, K. Hattori, T. Hattori and Watanabe [9] studied the asymptotically one-dimensional diffusion processes on the SG (see also Hambly and Jones [10], Hambly and Yang [13]). The random walk they considered is in fact the normalized probability of (a n , b n , b n ) as transition probability on the three sides of the n-level cells of the SG.…”