The Exact Hausdorff Measure Function of the Level Sets of Multi–parameter Symmetric Stable Process

In this paper, we investigate the Hausdorff measure for level sets of N–parameter Rd–valued stable processes, and develop a means of seeking the exact Hausdorff measure function for level sets of N–parameter Rd–valued stable processes. We show that the exact Hausdorff measure function of level sets of N–parameter Rd–valued symmetric stable processes of index α is ϕ(r) = rN−d/α(log log 1/r)d/α when Nα > d. In addition, we obtain a sharp lower bound for the Hausdorff measure of level sets of general (N, d,α) strictly stable processes.