Sharp non-asymptotic Concentration Inequalities for the Approximation of the Invariant Measure of a Diffusion

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.