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 $2$ 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.
Keywords: velocity jump process, Langevin diffusion, piecewise deterministic Markov process, molecular dynamics, exponential ergodicity, hypocoercivity, Talay–Tubaro expansion, weak discretization error
Nicolaï Gouraud 1; Lucas Journel 2; Pierre Monmarché 3
@article{SMAI-JCM_2025__11__533_0,
author = {Nicola{\"\i} Gouraud and Lucas Journel and Pierre Monmarch\'e},
title = {The velocity jump {Langevin} process and its splitting scheme: long time convergence and numerical accuracy},
journal = {The SMAI Journal of computational mathematics},
pages = {533--584},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {11},
year = {2025},
doi = {10.5802/smai-jcm.134},
language = {en},
url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.134/}
}
TY - JOUR AU - Nicolaï Gouraud AU - Lucas Journel AU - Pierre Monmarché TI - The velocity jump Langevin process and its splitting scheme: long time convergence and numerical accuracy JO - The SMAI Journal of computational mathematics PY - 2025 SP - 533 EP - 584 VL - 11 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.134/ DO - 10.5802/smai-jcm.134 LA - en ID - SMAI-JCM_2025__11__533_0 ER -
%0 Journal Article %A Nicolaï Gouraud %A Lucas Journel %A Pierre Monmarché %T The velocity jump Langevin process and its splitting scheme: long time convergence and numerical accuracy %J The SMAI Journal of computational mathematics %D 2025 %P 533-584 %V 11 %I Société de Mathématiques Appliquées et Industrielles %U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.134/ %R 10.5802/smai-jcm.134 %G en %F SMAI-JCM_2025__11__533_0
Nicolaï Gouraud; Lucas Journel; Pierre Monmarché. The velocity jump Langevin process and its splitting scheme: long time convergence and numerical accuracy. The SMAI Journal of computational mathematics, Volume 11 (2025), pp. 533-584. doi: 10.5802/smai-jcm.134
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