Exact targeting of Gibbs distributions using velocity-jump processes

TL;DR

This paper introduces a new velocity jump process that allows for exact simulation of Gibbs distributions, converges to Langevin or Hamiltonian dynamics as the time step shrinks, and guarantees precise stationary distribution matching the target.

Contribution

It presents a novel velocity jump process that is exactly simulatable, converges to classical dynamics, and maintains the target distribution without velocity reflections.

Findings

Process converges to Langevin/Hamiltonian dynamics as time step approaches zero.

Stationary distribution is exactly the target Gibbs distribution.

Numerical experiments demonstrate the method's effectiveness.

Abstract

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.