  We consider a stochastic perturbation of the $\alpha$-Navier-Stokes model.
The stochastic perturbation is an additive space-time noise of trace class.
Under a natural condition about the trace of operator $Q$ in front of the
noise, we prove the existence and uniqueness of strong solution, continuous in
time in classical spaces of $L^{2}$ functions with estimates of non-linear
terms. It is based on a priori estimate of solutions of finite-dimensional
systems, and tightness of the approximated solution.
  Moreover, by studying the derivative of the solution with respect to the
initial data, we can prove exponential moment of the approximated solutions,
which is enough to obtain Strong Feller property and irreducibility of the
transition semigroup. This leads naturally to the existence and uniqueness of
an invariant measure.
