  In this paper we provide performance guarantees for hypocoercive
non-reversible MCMC samplers $X_t$ with invariant measure $\mu^*$ and our
results apply in particular to the Langevin equation, Hamiltonian Monte-Carlo,
and the bouncy particle and zig-zag samplers. Specifically, we establish a
concentration inequality of Bernstein type for ergodic averages $\frac{1}{T}
\int_0^T f(X_t)\, dt$. As a consequence we provide performance guarantees: (a)
explicit non-asymptotic confidence intervals for $\int f d\mu^*$ when using a
finite time ergodic average with given initial condition $\mu$ and (b)
uncertainty quantification bounds, expressed in terms of relative entropy rate,
on the bias of $\int f d\mu^*$ when using an alternative or approximate
processes $\widetilde{X}_t$. (Results in (b) generalize recent results
(arXiv:1812.05174) from the authors for coercive dynamics.) The concentration
inequality is proved by combining the approach via Feynmann-Kac semigroups
first noted by Wu with the hypocoercive estimates of Dolbeault, Mouhot and
Schmeiser (arXiv:1005.1495) developed for the Langevin equation and recently
generalized to partially deterministic Markov processes by Andrieu et al.
(arXiv:1808.08592)
