Sharp non-asymptotic concentration inequalities for the approximation of the invariant distribution of a diffusion

Let ${\left(Y_t\right)}_{t\ge 0}$ be an ergodic diffusion with invariant distribution $\nu$. Consider the empirical measure ${\nu }_n?{\left({\sum }_{k=1}^n{\gamma }_k\right)}^{?1}{\sum }_{k=1}^n{\gamma }_k{\delta }_{X_{k?1}}$ where ${\left(X_k\right)}_{k\ge 0}$ is an Euler scheme with decreasing steps ${\left({\gamma }_k\right)}_{k\ge 0}$ which approximates ${\left(Y_t\right)}_{t\ge 0}$. Given a test function $f$, we obtain sharp concentration inequalities for ${\nu }_n\left(f\right)?\nu \left(f\right)$ which improve the results in Honoré et al. (2019). Our hypotheses on the test function $f$ cover many real applications: either $f$ is supposed to be a coboundary of the infinitesimal generator of the diffusion, or $f$ is supposed to be Lipschitz.