Degree two ergodic theorem for divergence-free stationary random fields

Abstract

We prove the ergodic theorem for surface integrals of divergence-free stationary random fields of ℝ3. Mean convergence in $$ \mathbb{L}^p $$ spaces takes place as soon as the field is $$ \mathbb{L}^p $$ -integrable. The condition of integrability for the pointwise convergence is expressed by a Lorentz norm. This theorem is an ergodic theorem for cocycles of degree 2, analogous to the ergodic theorem for cocycles of degree 1 proved in [1].