  We introduce a novel class of generative models based on piecewise
deterministic Markov processes (PDMPs), a family of non-diffusive stochastic
processes consisting of deterministic motion and random jumps at random times.
Similarly to diffusions, such Markov processes admit time reversals that turn
out to be PDMPs as well. We apply this observation to three PDMPs considered in
the literature: the Zig-Zag process, Bouncy Particle Sampler, and Randomised
Hamiltonian Monte Carlo. For these three particular instances, we show that the
jump rates and kernels of the corresponding time reversals admit explicit
expressions depending on some conditional densities of the PDMP under
consideration before and after a jump. Based on these results, we propose
efficient training procedures to learn these characteristics and consider
methods to approximately simulate the reverse process. Finally, we provide
bounds in the total variation distance between the data distribution and the
resulting distribution of our model in the case where the base distribution is
the standard $d$-dimensional Gaussian distribution. Promising numerical
simulations support further investigations into this class of models.
