Fractional Gaussian fields on the Sierpinski gasket and related fractals

Abstract

We define and study a fractional Gaussian field $X$ with Hurst parameter $H$ on the Sierpiński 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 Sierpiński carpet.