Large deviations for the empirical measure of the zig-zag process
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The zig-zag process is a piecewise deterministic Markov process in position and velocity space. The process can be designed to have an arbitrary marginal probability density for its position coordinate, which makes it suitable for the numerical simulation of general 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 encompassing the zig-zag process, allowing us to 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 a more explicit expression for the Donsker-Varadhan functional for the case of a compact state space.
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