Local scaling limits of Levy driven fractional random fields

We obtain a complete description of local anisotropic scaling limits for a class of fractional random fields X on R-2 written as stochastic integral with respect to infinitely divisible random measure. The scaling procedure involves increments of X over points the distance between which in the horizontal and vertical directions shrinks as O(lambda) and O(lambda(gamma)) respectively as lambda down arrow 0, for some gamma > 0. We consider two types of increments of X: usual increment and rectangular increment, leading to the respective concepts of gamma-tangent and gamma-rectangent random fields. We prove that for above X both types of local scaling limits exist for any gamma > 0 and undergo a transition, being independent of gamma > gamma(0) and gamma < gamma(0), for some gamma(0) > 0; moreover, the 'unbalanced' scaling limits (gamma not equal gamma(0)) are (H-1, H-2)-multi self-similar with one of H-i, i = 1, 2, equal to 0 or 1. The paper extends Pilipauskaite and Surgailis (Stochastic Process. Appl. 127 (2017) 2751-2779) and Surgailis (Stochastic Process. Appl. 130 (2020) 7518-7546) on largescale anisotropic scaling of random fields on Z(2) and Benassi et al. (Bernoulli 10 (2004) 357-373) on 1-tangent limits of isotropic fractional Levy random fields.