Invariant measure of duplicated diffusions and application to Richardson-Romberg extrapolation
Vincent Lemaire, Gilles Pagès, Fabien Panloup · 2013
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 mainly focus on the interesting case where the two trajectories follow the same Brownian path. In this case, we first show that uniqueness is essentially always true in the one-dimensional case. Then, in the multidimensional case, we build some explicit counter-examples where the uniqueness property is not satisfied and then, give explicit conditions on the drift and diffusion coefficient functions (which can be interpreted as the negativity of a non-infinitesimal Lyapunov exponent associated with the dynamical system) to obtain uniqueness for invariant distribution of the duplicated system. As a main application, we investigate the Richardson-Romberg extrapolation for the numerical approximation of the invariant measure of the initial ergodic Brownian diffusion.