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

TL;DR

This paper investigates the small-scale behaviour of non-fixed point diffusions on fractals, revealing a natural limit process supported on a different fractal obtained via a Gromov-Hausdorff limit, and establishes weak convergence results.

Contribution

It introduces a framework for understanding the microscale limits of non-fixed point diffusions on fractals, extending previous work on the Sierpinski gasket.

Findings

Identification of a limit diffusion supported on a different fractal

Proof of weak convergence of rescaled diffusions

Answering Hattori's question on the ultraviolet limit

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 Hattori (1994) about the ultraviolet limit of the asymptotically one-dimensional diffusion process on the Sierpinski gasket.