Conformal Walk Dimension: Its Universal Value and the Non-attainment for the Sierpinski Carpet

It is an established result in the field of analysis of diffusion processes on fractals, that the transition density of the diffusion typically satisfies analogs of Gaussian bounds which involve a space-time scaling exponent β greater than two and thereby are called SUB-Gaussian bounds. The exponent β, called the walk dimension of the diffusion, could be considered as representing “how close the geometry of the fractal is to being smooth”. It has been observed by Kigami in [Math. Ann. 340 (2008), 781-804] that, in the case of the standard two-dimensional Sierpinski gasket, one can decrease this exponent to two (so that Gaussian bounds hold) by suitable changes of the metric and the measure while keeping the associated Dirichlet form (the quadratic energy functional) the same. Then it is natural to ask how general this phenomenon is for diffusions.

This talk is aimed at presenting (partial) answers to this question. More specifically, the talk will present the following results:

(1) For any symmetric diffusion on a locally compact separable metric measure space in which any bounded set is relatively compact, the infimum over all possible values of the exponent β after “suitable” changes of the metric and the measure is ALWAYS two unless it is infinite. (We call this infimum the conformal walk dimension of the diffusion).

(2) The infimum as in (1) above is NOT attained, in the case of the Brownian motion on the standard (two-dimensional) Sierpinski carpet (as well as that on the standard three-and higher-dimensional Sierpinski gaskets).

This talk is based on joint works with Mathav Murugan (UBC). The results are given in arXiv:2008.12836, except for the non-attainment result for the Sierpinski carpet in (2) above, which is in progress.