For piecewise deterministic samplers such as Randomized Hamiltonian Monte Carlo (RHMC), Bouncy Particle Sampler (BPS) or Zig-Zag Process (ZZP), long-time exponential convergence rates have been established in previous works using Harris or $L^2$ hypocoercivity approaches. In particular, in the $L^2$ framework, a so-called \emph{diffusive-to-ballistic} speedup was known for log-concave targets, according to which the convergence rates of these samplers, with suitable parameters, are quadratically improved with respect to the standard overdamped Langevin diffusion process. A recent work by Jianfeng Lu showed that this speedup also holds for the kinetic Langevin diffusion process when the convergence is stated in terms of relative entropy, raising the question whether this also holds for piecewise deterministic samplers. The present work provides a positive and a negative answer to this: first, we show that the speedup holds in entropy for RHMC; second, we show that for BPS or ZZS, even for a standard Gaussian target, a similar result cannot hold, and even that exponential convergence (at any rate) in entropy fails.
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