Continuity in the Hurst Index of the Local Times of Anisotropic Gaussian Random Fields

Let {{XH(t), t ∈ RN}, H ∈ (0, 1)N} be a family of (N, d)-anisotropic Gaussian random fields with generalized Hurst indices H = (H1, . . . ,HN ) ∈ (0, 1) . Under certain general conditions, we prove that the local time of {XH0(t), t ∈ RN} is jointly continuous whenever ∑N `=1 1/H 0 ` > d. Moreover we show that, when H approaches H , the law of the local times of X(t) converges weakly [in the space of continuous functions] to that of the local time of X 0 . The latter theorem generalizes the result of Jolis and Viles (2007) for one-parameter fractional Brownian motions with values in R to a wide class of (N, d)Gaussian random fields. The main argument of this paper relies on the recently developed sectorial local nondeterminism for anisotropic Gaussian random fields. Running head: Continuity in the Hurst index of the local times of Gaussian fields 2000 AMS Classification numbers: 60G15, 60G17; 42C40; 28A80.