Eddy viscosity by Lévy transport noises

Abstract

We consider stochastic linear transport equations in dimension $d\ge 2$ and stochastic 2D Euler equations with $L^2$-initial condition, both driven by Lévy transport noise in the Marcus sense. Under a suitable scaling limit of the noises, we prove that the solutions to the stochastic linear transport equations converge to the solution of the deterministic heat equation, while weak solutions to the stochastic 2D Euler equations converge weakly to the unique solution of the deterministic 2D Navier--Stokes equation. These results show that small-scale jump noises generate eddy viscosity, extending the recent studies on Itô--Stratonovich diffusion limits to the discontinuous setting.

arXiv comment43 pages. We have added Lem 2.9 and Prop 2.10 on properties of convergence in Skorohod topology, Prop 3.2 on the existence of stochastic flow of Marcus SDE. We have also rewritten the convergence of stochastic integral in Step 3 of proof of Theorem 1.5