Continuity in the Hurst index of the local times of anisotropic Gaussian random fields

Let be a family of (N,d)-anisotropic Gaussian random fields with generalized Hurst indices H=(H1,...,HN)[set membership, variant](0,1)N. Under certain general conditions, we prove that the local time of is jointly continuous whenever . Moreover we show that, when H approaches H0, the law of the local times of XH(t) converges weakly [in the space of continuous functions] to that of the local time of XH0. The latter theorem generalizes the result of [M. Jolis, N. Viles, Continuity in law with respect to the Hurst parameter of the local time of the fractional Brownian motion, J. Theoret. Probab. 20 (2007) 133-152] for one-parameter fractional Brownian motions with values in 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.