In this talk, we introduce fractional stable and Gaussian random fields on the Sierpinski gasket in the sense of distributions.

We first focus on the existence of a density with respect to the Hausdorff measure. When this density field exists, we then study the smoothness of its sample paths. In the Gaussian framework, such a field always admits a modification with Hölder continuous sample paths. However, roughly speaking, in the non-Gaussian framework, the sample paths cannot be smoother than the Riesz fractional kernel. As a consequence, in this case, the density field either has unbounded sample paths or admits a modification with Hölder continuous sample paths. In the stable framework, Hölder regularity follows from an upper bound on the modulus of continuity, which we obtain using a LePage series representation. Finally, the density field also satisfies certain scaling and invariance properties.

This talk is joint work with Fabrice Baudoin (Aarhus University).