  For an ergodic Brownian diffusion with invariant measure $\nu$, we consider a
sequence of empirical distributions ($\nu$n) n$\ge$1 associated with an
approximation scheme with decreasing time step ($\gamma$n) n$\ge$1 along an
adapted regular enough class of test functions f such that f --$\nu$(f) is a
coboundary of the infinitesimal generator A. Denote by $\sigma$ the diffusion
coefficient and $\Phi$ the solution of the Poisson equation A$\Phi$ = f --
$\nu$(f). When the square norm of |$\sigma$ * $\Phi$| 2 lies in the same
coboundary class as f , we establish sharp non-asymptotic concentration bounds
for suitable normalizations of $\nu$n(f) -- $\nu$(f). Our bounds are optimal in
the sense that they match the asymptotic limit obtained by Lamberton and
Pag{\`e}s in [LP02], for a certain large deviation regime. In particular, this
allows us to derive sharp non-asymptotic confidence intervals. We provide as
well a Slutsky like Theorem, for practical applications, where the deviation
bounds are also asymptotically independent of the corresponding Poisson
problem. Eventually, we are able to handle, up to an additional constraint on
the time steps, Lipschitz sources f in an appropriate non-degenerate setting.
