Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems

A classical damping Hamiltonian system perturbed by a random force is considered. The locally uniform large deviation principle of Donsker and Varadhan is established for its occupation empirical measures for large time, under the condition, roughly speaking, that the force driven by the potential grows infinitely at infinity. Under the weaker condition that this force remains greater than some positive constant at infinity, we show that the system converges to its equilibrium measure with exponential rate, and obeys moreover the moderate deviation principle. Those results are obtained by constructing appropriate Lyapunov test functions, and are based on some results about large and moderate deviations and exponential convergence for general strong-Feller Markov processes. Moreover, these conditions on the potential are shown to be sharp.