  The Bouncy Particle Sampler is a Markov chain Monte Carlo method based on a
nonreversible piecewise deterministic Markov process. In this scheme, a
particle explores the state space of interest by evolving according to a linear
dynamics which is altered by bouncing on the hyperplane tangent to the gradient
of the negative log-target density at the arrival times of an inhomogeneous
Poisson Process (PP) and by randomly perturbing its velocity at the arrival
times of an homogeneous PP. Under regularity conditions, we show here that the
process corresponding to the first component of the particle and its
corresponding velocity converges weakly towards a Randomized Hamiltonian Monte
Carlo (RHMC) process as the dimension of the ambient space goes to infinity.
RHMC is another piecewise deterministic non-reversible Markov process where a
Hamiltonian dynamics is altered at the arrival times of a homogeneous PP by
randomly perturbing the momentum component. We then establish dimension-free
convergence rates for RHMC for strongly log-concave targets with bounded
Hessians using coupling ideas and hypocoercivity techniques.
