Invariant measures for stochastic 3D Lagrangian-averaged Navier–Stokes equations with infinite delay

In this paper we investigate stochastic dynamics and invariant measures for 3D Lagrangian-averaged Navier–Stokes (LANS) equations driven by infinite delay additive noise. We first use Galerkin approximations, a priori estimates the standard Gronwall lemma to show well-posedness corresponding random equation, whose solution operators generate dynamical system. Next, asymptotic compactness system is established via Ascoli–Arzelà theorem. Besides, derive existence of global attractor Moreover, prove that bounded continuous with respect initial values. Eventually, construct family Borel probability measures, which supported attractor.