Inviscid Limit of the Stochastic Hyperviscous Navier-Stokes Equations and Invariant Measures for the Euler Equations in ${\mathbb{R}}^2$

We prove the existence and some moment estimates for an invariant measure $\mu$ for the two-dimensional ($2$D) deterministic Euler equations on the unbounded domain ${\mathbb{R}}^2$ and with highly regular initial data. The result is achieved by first showing the existence of Markov stationary processes which solve the hyperviscous $2$D Navier-Stokes equations with kinematic viscosity $\nu >0$ and an additive stochastic noise scaling as $\sqrt{\nu }$. We then study the inviscid limit and prove that, as $\nu$ tends to $0$, these processes converge, in an appropriate trajectory space, to a pathwise stationary solution to the Euler equations. Its law is the sought invariant measure $\mu$.