Limit motion of an Ornstein–Uhlenbeck particle on the equilibrium manifold of a force field

In this paper we consider a position–velocity Ornstein–Uhlenbeck process in an external gradient force field pushing it toward a smoothly imbedded submanifold of ℝ^d, ℳ. The force is chosen so that ℳ is asymptotically stable for the associated deterministic flow. We examine the asymptotic behavior of the system when the force intensity diverges together with the diffusion and the damping coefficients, with appropriate speed. We prove that, under some natural conditions on the initial data, the sequence of position processes is relatively compact, any limit process is constrained on ℳ, and satisfies an explicit stochastic differential equation which, for compact ℳ, has a unique solution.