  This work introduces and studies a new family of velocity jump Markov
processes directly amenable to exact simulation with the following two
properties: i) trajectories converge in law when a time-step parameter vanishes
towards a given Langevin or Hamil-tonian dynamics; ii) the stationary
distribution of the process is always exactly given by the product of a
Gaussian (for velocities) by any target log-density whose gradient is pointwise
computabe together with some additional explicit appropriate upper bound. The
process does not exhibit any velocity reflections (jump sizes can be
controlled) and is suitable for the 'factorization method'. We provide a
rigorous mathematical proof of: i) the small time-step convergence towards
Hamiltonian/Langevin dynamics, as well as ii) the exponentially fast
convergence towards the target distribution when suitable noise on velocity is
present. Numerical implementation is detailed and illustrated.
