Hitting probabilities and the Hausdorff dimension of the inverse images of a class of anisotropic random fields

Let X = {X(t): t . RN} be an anisotropic random field with values in Rd. Under certain conditions on X, we establish upper and lower bounds on the hitting probabilities of X in terms of respectively Hausdorff measure and Bessel.Riesz capacity. We also obtain the Hausdorff dimension of its inverse image, and the Hausdorff and packing dimensions of its level sets. These results are applicable to non-linear solutions of stochastic heat equations driven by a white in time and spatially homogeneous Gaussian noise and anisotropic Guassian random fields.