We examine the paths of the stable fields that are the analogs of index-$\beta$ Gaussian fields. We find Holder conditions on their paths and find the Hausdorff dimension of the image, graph and level sets when we have local nondeterminism, generalizing the Gaussian results.
"Path Properties of Index-$\beta$ Stable Fields." Ann. Probab. 16 (4) 1596 - 1607, October, 1988. https://doi.org/10.1214/aop/1176991586