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.
J. Bierkens acknowledges support by the Dutch Research Council (NWO) for the research project Zig-zagging through computational barriers with project number 016.Vidi.189.043. P. Nyquist acknowledges funding from the NWO Grant 613.009.101 at the outset of this work. During the final stage P. Nyquist was IBM Visiting Professor in the Division of Applied Mathematics at Brown University, and the hospitality and financial support of the Division is gratefully acknowledged. M. Schlottke acknowledges financial support through NWO Grant 613.001.552.
The authors thank Jin Feng for several helpful discussions on topics directly related to this paper. We acknowledge constructive comments by the anonymous referee, which have helped to improve the clarity of this paper.
"Large deviations for the empirical measure of the zig-zag process." Ann. Appl. Probab. 31 (6) 2811 - 2843, December 2021. https://doi.org/10.1214/21-AAP1663