Invariant measures for non-autonomous dissipative dynamical systems

Given a non-autonomous process $U(\cdot,\cdot)$ on a complete separable metric space $X$ that has a pullback attractor $A(\cdot)$, we construct a family of invariant Borel probability measures $\{\mu_t\}_{t\in \mathbb{R}}$: the measures satisfy ${\rm supp }\,{\mu_t}\subset A(t)$ for all $t\in \mathbb{R}$ and the invariance property $\mu_t(E)=\mu_\tau(U(t,\tau)^{-1}E)$ for every Borel set $E\in X$. Our construction uses the generalised Banach limit. We then show that a Liouville-type equation holds for the evolution of $\mu_t$ under the process $U(\cdot,\cdot)$ generated by the ordinary differential equation $u_t=F(t,u)$ on a Banach space, and apply our theory to the non-autonomous 2D Navier--Stokes equations on unbounded domains satisfying a Poincare inequality.