Fractional stable random fields on the Sierpiński gasket

We define and study fractional stable random fields on the Sierpiński gasket. Such fields are formally defined as $(-\mathrm{\Delta }{)}^{-s}{W}_{K,\alpha }$, where $\mathrm{\Delta }$ is the Laplace operator on the gasket and $W_{K,\alpha }$ is a stable random measure. Both Neumann and Dirichlet boundary conditions for $\mathrm{\Delta }$ are considered. Sample paths regularity and scaling properties are obtained. The techniques we develop are general and extend to the more general setting of the Barlow fractional spaces.