  We study Langevin dynamics with a kinetic energy different from the standard,
quadratic one in order to accelerate the sampling of Boltzmann-Gibbs
distributions. In particular, this kinetic energy can be non-globally
Lipschitz, which raises issues for the stability of discretizations of the
associated Langevin dynamics. We first prove the exponential convergence of the
law of the continuous process to the Boltzmann-Gibbs measure by a hypocoercive
approach, and characterize the asymptotic variance of empirical averages over
trajectories. We next develop numerical schemes which are stable and of weak
order two, by considering splitting strategies where the discretizations of the
fluctuation/dissipation are corrected by a Metropolis procedure. We use the
newly developped schemes for two applications: optimizing the shape of the
kinetic energy for the so-called adaptively restrained Langevin dynamics (which
considers perturbations of standard quadratic kinetic energies vanishing around
the origin); and reducing the metastability of some toy models using
non-globally Lipschitz kinetic energies.
