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

Hattori, Tetsuya; Watanabe, Hiroshi

doi:10.1007/bf02508466

Cited by 2 publications

(1 citation statement)

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“…In [15] it is observed that, even in the situation where there is no fixed point for the renormalization map, this procedure could lead to a diffusion on the O(1) scale which was non-degenerate even though at both small and large scale the diffusion is degenerate. This was illustrated in the case of abc-gaskets where the ratios between a, b and c are such that there is no fixed point [16].…”

Section: Introductionmentioning

confidence: 99%

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

Hambly

Yang

2018

J. Fractal Geom.

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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.

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Section: Introductionmentioning

confidence: 99%