The Langevin dynamics is a diffusion process extensively used, in particular in molecular dynamics simu-lations, to sample Gibbs measures. Some alternatives based on (piecewise deterministic) kinetic velocity jumpprocesses 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, anumerical scheme based on hybrid dynamics combining velocity jumps and Langevin diffusion, numericallymore efficient than their classical Langevin counterparts, has been introduced for computational chemistry in [42]. The present work is devoted to the numerical analysis of this scheme. Our main results are, first, theexponential ergodicity of the continuous-time velocity jump Langevin process, second, a Talay-Tubaro expan-sion of the invariant measure of the numerical scheme, showing in particular that the scheme is of weak order2 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, newdifficulties arise from the jump operator, which is non-local.

翻译：朗之万动力学是一种扩散过程，在分子动力学模拟等领域被广泛用于采样吉布斯测度。基于（分段确定性）动力学速度跳跃过程的替代方案在过去十年中受到关注。后者的一个优势在于能够在连续时间层面分裂力场，从而降低轨迹采样的数值成本。受此启发，文献[42]针对计算化学领域提出了一种结合速度跳跃与朗之万扩散的混合动力学数值格式，其数值效率优于经典朗之万方法。本文致力于对该格式进行数值分析。我们的主要成果包括：首先证明连续时间速度跳跃朗之万过程的指数遍历性；其次通过Talay-Tubaro展开分析数值格式不变测度的性质，特别证明该格式在步长上具有二阶弱收敛阶；最后给出相应实用MCMC估计量（可结合Richardson外推法）二次风险的界值。相较于以往关于朗之万扩散的研究，非局域的跳跃算子带来了新的理论挑战。