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

Let X = { X ( t ), t ∈ ℝ N } be a Gaussian random field with values in ℝ d defined by X ( t ) = ( X 1 ( t ), …, X d ( t )), where X 1 , …, X d are independent copies of a centered Gaussian random field X 0 . Under certain general conditions on X 0 , we study the hitting probabilities of X and determine the Hausdorff dimension of the inverse image X −1 ( F ), where F ⊆ ℝ d is a non‐random Borel set. The class of Gaussian random fields that satisfy our conditions includes not only fractional Brownian motion and the Brownian sheet, but also such anisotropic fields as fractional Brownian sheets, solutions to stochastic heat equation driven by space‐time white noise and the operator‐scaling Gaussian random fields with stationary increments constructed in [H. Biermé, M. M. Meerschaert and H.‐P. Scheffler, ‘Operator scaling stable random fields’, Stochastic Process. Appl. 117 (2007) 312–332.].