Local Limit Theorems for Energy Fluxes of Infinite Divisible Random Fields

Abstract: We study the local asymptotic behavior of divergence-like functionals of a family of $d$-dimensional Infinitely Divisible Random Fields. Specifically, we derive limit theorems of surface integrals over Lipschitz manifolds for this class of fields when the region of integration shrinks to a single point. We show that in most cases, convergence stably in distribution holds after a proper normalization. Furthermore, the limit random fields can be described in terms of stochastic integrals with respect to a L\'evy basis. We additionally discuss how our results can be used to measure the kinetic energy of a possibly turbulent flow.