Hitting Probability and the Hausdorff Measure of the Level sets for Spherical Gaussian Fields

Consider an isotropic spherical Gaussian random field T with values in R^{d}. We investigate two problems: (i) When is the level set T^{-1}(t) nonempty with positive probability for any t\in R^{d} ? (ii) If the level set is nonempty, what is its Hausdorff measure? These two question are not only very important in potential theory for random fields, but also foundamental in geometric measure theory. We give a complete answer to the questions under some very mild conditions.