In this work, we establish $\mathrm{L}^2$-exponential convergence for a broad
class of Piecewise Deterministic Markov Processes recently proposed in the
context of Markov Process Monte Carlo methods and covering in particular the
Randomized Hamiltonian Monte Carlo, the Zig-Zag process and the Bouncy Particle
Sampler. The kernel of the symmetric part of the generator of such processes is
non-trivial, and we follow the ideas recently introduced by (Dolbeault et al.,
2009, 2015) to develop a rigorous framework for hypocoercivity in a fairly
general and unifying set-up, while deriving tractable estimates of the
constants involved in terms of the parameters of the dynamics. As a by-product
we characterize the scaling properties of these algorithms with respect to the
dimension of classes of problems, therefore providing some theoretical evidence
to support their practical relevance.

本文研究基于分段确定性马尔科夫过程的随机哈密顿蒙特卡洛、Zig-Zag 过程和 Bouncy Particle 采样等一系列过程的 L2 指数收敛性和低效性，证明这些算法在高维问题中具有较好的可行性。

分段确定性马尔可夫过程蒙特卡罗的次弱收敛性

Hypocoercivity of Piecewise Deterministic Markov Process-Monte Carlo

The Bouncy Particle Sampler is a Markov chain Monte Carlo method based on a
nonreversible piecewise deterministic Markov process. In this scheme, a
particle explores the state space of interest by evolving according to a linear
dynamics which is altered by bouncing on the hyperplane tangent to the gradient
of the negative log-target density at the arrival times of an inhomogeneous
Poisson Process (PP) and by randomly perturbing its velocity at the arrival
times of an homogeneous PP. Under regularity conditions, we show here that the
process corresponding to the first component of the particle and its
corresponding velocity converges weakly towards a Randomized Hamiltonian Monte
Carlo (RHMC) process as the dimension of the ambient space goes to infinity.
RHMC is another piecewise deterministic non-reversible Markov process where a
Hamiltonian dynamics is altered at the arrival times of a homogeneous PP by
randomly perturbing the momentum component. We then establish dimension-free
convergence rates for RHMC for strongly log-concave targets with bounded
Hessians using coupling ideas and hypocoercivity techniques.

本文提出基于非可逆分段确定性马尔可夫过程的 Bouncy Particle Sampler 算法，通过在均匀和不均匀 Poisson 过程到达时弹跳并随机扰动速度，使粒子探测兴趣状态空间。经过充分的正则性条件分析，本文证明了该算法的一部分和其相应的速度在空间维度趋于无穷时弱收敛于随机哈密顿蒙特卡罗 (RHMC) 算法。通过耦合思想和 Hypocoercivity 技术，我们还建立了 RHMC 算法在具有有界 Hessian 的强对数凹目标上的无维收敛速率。