  We study the exponential convergence to the stationary state for
nonequilibrium Langevin dynamics, by a perturbative approach based on
hypocoercive techniques developed for equilibrium Langevin dynamics. The
Hamiltonian and overdamped limits (corresponding respectively to frictions
going to zero or infinity) are carefully investigated. In particular, the
maximal magnitude of admissible perturbations are quantified as a function of
the friction. Numerical results based on a Galerkin discretization of the
generator of the dynamics confirm the theoretical lower bounds on the spectral
gap.
