With a view to numerical applications we address the following question: given an ergodic Brownian diffusion with a unique invariant distribution, what are the invariant distributions of the duplicated system consisting of two trajectories? We mainly focus on the interesting case where the two trajectories are driven by the same Brownian path. Under this assumption, we first show that uniqueness is essentially always true in the one-dimensional case. In the multidimensional case, we begin by building some explicit counter-examples where the uniqueness property is not fulfilled. Then, we provide a series of criterions depending on the drift and diffusion coefficients (which can be viewed as the negativity of a \textit{non-infinitesimal Lyapunov exponent} associated with the dynamical system) under which uniqueness of the invariant distribution of the duplicated system (weak confluence) or pathwise confluence hold. As examples, we apply our criterions to some non-trivially confluent settings such as classes of gradient systems with non-convex potentials or diffusions where the confluence is generated by the diffusive component. As a main application, we investigate the Richardson-Romberg extrapolation for the numerical approximation of the invariant measure of the initial ergodic Brownian diffusion.