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

Let {{XH(t),t∈RN},H∈(0,1)N}{{XH(t),t∈RN},H∈(0,1)N} be a family of (N,d)(N,d)-anisotropic Gaussian random fields with generalized Hurst indices H=(H1,…,HN)∈(0,1)NH=(H1,…,HN)∈(0,1)N. Under certain general conditions, we prove that the local time of {XH0(t),t∈RN}{XH0(t),t∈RN} is jointly continuous whenever ∑ℓ=1N1/Hℓ0>d. Moreover we show that, when HH approaches H0H0, the law of the local times of XH(t)XH(t) converges weakly [in the space of continuous functions] to that of the local time of XH0XH0. 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 RR to a wide class of (N,d)(N,d)-Gaussian random fields. The main argument of this paper relies on the recently developed sectorial local nondeterminism for anisotropic Gaussian random fields.