  We obtain non-asymptotic Gaussian concentration bounds for the difference
between the invariant measure $\nu$ of an ergodic Brownian diffusion process
and the empirical distribution of an approximating scheme with decreasing time
step along a suitable class of (smooth enough) test functions f such that f --
$\nu$(f) is a coboundary of the infinitesimal generator. We show that these
bounds can still be improved when the (squared) Fr{\"o}benius norm of the
diffusion coefficient lies in this class. We apply these bounds to design
computable non-asymptotic confidence intervals for the approximating scheme. As
a theoretical application, we finally derive non-asymptotic deviation bounds
for the almost sure Central Limit Theorem.
