Invariant measures for non-autonomous dissipative dynamical systems

Given a non-autonomous process U(·,·) on a complete separable metric space X that has a pullback attractor A(·), we construct a family of invariant Borel probability measures \μₜ∈ R: the measures satisfy supp \,μₜ⊂ A(t) for all t∈ R and the invariance propertyμₜ(E)=μ_τ(U(t,τ)⁻¹E) for every Borel set E∈ X. Our construction uses the generalised Banach limit. We then show that a Liouville-type equation holds for the evolution of μₜ under the process U(·,·) generated by the ordinary differential equation uₜ=F(t,u) on a Banach space, and apply our theory to the non-autonomous 2D Navier--Stokes equations on unbounded domains satisfying a Poincaré inequality.