The velocity jump Langevin process and its splitting scheme: long time convergence and numerical accuracy

The Langevin dynamics is a diffusion process extensively used, in particular in molecular dynamics simulations, to sample Gibbs measures. Some alternatives based on (piecewise deterministic) kinetic velocity jump processes have gained interest over the last decade. One interest of the latter is the possibility to split forces (at the continuous-time level), reducing the numerical cost for sampling the trajectory. Motivated by this, a numerical scheme based on hybrid dynamics combining velocity jumps and Langevin diffusion, numerically more efficient than their classical Langevin counterparts, has been introduced for computational chemistry in [43]. The present work is devoted to the numerical analysis of this scheme. Our main results are, first, the exponential ergodicity of the continuous-time velocity jump Langevin process, second, a Talay–Tubaro expansion of the invariant measure of the numerical scheme on the torus, showing in particular that the scheme is of weak order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>2</mml:mn> </mml:math> in the step-size and, third, a bound on the quadratic risk of the corresponding practical MCMC estimator (possibly with Richardson extrapolation). With respect to previous works on the Langevin diffusion, new difficulties arise from the jump operator, which is non-local.