  The zig-zag process is a piecewise deterministic Markov process in position
and velocity space. The process can be designed to have an arbitrary Gibbs type
marginal probability density for its position coordinate, which makes it
suitable for Monte Carlo simulation of continuous probability distributions. An
important question in assessing the efficiency of this method is how fast the
empirical measure converges to the stationary distribution of the process. In
this paper we provide a partial answer to this question by characterizing the
large deviations of the empirical measure from the stationary distribution.
Based on the Feng-Kurtz approach, we develop an abstract framework aimed at
encompassing piecewise deterministic Markov processes in position-velocity
space. We derive explicit conditions for the zig-zag process to allow the
Donsker-Varadhan variational formulation of the rate function, both for a
compact setting (the torus) and one-dimensional Euclidean space. Finally we
derive an explicit expression for the Donsker-Varadhan functional for the case
of a compact state space and use this form of the rate function to address a
key question concerning the optimal choice of the switching rate of the zig-zag
process.
