  Zig-Zag is Piecewise Deterministic Markov Process, efficiently used for
simulation in an MCMC setting. As we show in this article, it fails to be
exponentially ergodic on heavy tailed target distributions. We introduce an
extension of the Zig-Zag process by allowing the process to move with a
non-constant speed function $s$, depending on the current state of the process.
We call this process Speed Up Zig-Zag (SUZZ). We provide conditions that
guarantee stability properties for the SUZZ process, including non-explosivity,
exponential ergodicity in heavy tailed targets and central limit theorem.
Interestingly, we find that using speed functions that induce explosive
deterministic dynamics may lead to stable algorithms that can even mix faster.
We further discuss the choice of an efficient speed function by providing an
efficiency criterion for the one-dimensional process and we support our
findings with simulation results.
