Fractional Gaussian fields on the Sierpinski gasket and related fractals

TL;DR

This paper introduces fractional Gaussian fields on the Sierpiński gasket and similar fractals, constructed as solutions to fractional Laplacian equations driven by Gaussian white noise, extending the theory to various fractal structures.

Contribution

It defines and analyzes fractional Gaussian fields on fractals, providing a new framework for understanding Gaussian processes on complex geometric structures.

Findings

Defined fractional Gaussian fields on the Sierpiński gasket.

Extended the construction to other fractals like the Sierpiński carpet.

Connected the fields to fractional Laplacian equations driven by white noise.

Abstract

We define and study a fractional Gaussian field $X$ with Hurst parameter $H$ on the Sierpi\'nski gasket $K$ equipped with its Hausdorff measure $\mu$. It appears as a solution, in a weak sense, of the equation $(-\Delta)^s X =W$ where $W$ is a Gaussian white noise on $L_0^2(K,\mu)$, $\Delta$ the Laplacian on $K$ and $s= \frac{d_h+2H}{2d_w}$, where $d_h$ is the Hausdorff dimension of $K$ and $d_w$ its walk dimension. The construction of those fields is then extended to other fractals including the Sierpi\'nski carpet.