With a view to numerical applications we address the following question: given an ergodic Brownian diffusion with an unique invariant measure, what are the invariant measures of the duplicated system consisting of two trajectories ? We focus on the interesting case where the two trajectories follow the same Brownian path, and give explicit conditions on the drift and diffusion coefficient function to obtain uniqueness for invariant distribution of the duplicated system. As an application, we investigate the Richardson-Romberg extrapolation for the numerical approximation of the invariant measure of the initial ergodic Brownian diffusion.