Fractional Gaussian fields on the Sierpinski gasket and related fractals

We define and study a fractional Gaussian field $X$ with Hurst parameter $H$ on the Sierpiński gasket $K$ equipped with its Hausdorff measure $μ$. It appears as a solution, in a weak sense, of the equation $(-Δ)^s X =W$ where $W$ is a Gaussian white noise on $L_0^2(K,μ)$, $Δ$ 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.

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