Periodic homogenization for hypoelliptic diffusions
The long time behavior of an Ornstein-Uhlenbeck process is investigated, under the influence of a periodic drift. That involves the problem of homogenization for periodic hypoelliptic diffusions. It is proved that the rescaled particle position weakly converges to a Brownian motion, whose covariance matrix is determined by a solution of the Poisson equation. Estimates on the particle velocity are established. Upper bounds on the convergence rate in several norms were obtained and an optimality of the \(p\)-Wassserstein metric was indicated.
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