Piecewise deterministic Markov processes (PDMPs) are a type of continuous-time Markov process that combine deterministic flows with jumps. Recently, PDMPs have garnered attention within the Monte Carlo community as a potential alternative to traditional Markov chain Monte Carlo (MCMC) methods. The Zig-Zag sampler and the Bouncy particle sampler are commonly used examples of the PDMP methodology which have also yielded impressive theoretical properties, but little is known about their robustness to extreme dependence or isotropy of the target density. It turns out that PDMPs may suffer from poor mixing due to anisotropy and this paper investigates this effect in detail in the stylised but important Gaussian case. To this end, we employ a multi-scale analysis framework in this paper. Our results show that when the Gaussian target distribution has two scales, of order $1$ and $\epsilon$, the computational cost of the Bouncy particle sampler is of order $\epsilon^{-1}$, and the computational cost of the Zig-Zag sampler is either $\epsilon^{-1}$ or $\epsilon^{-2}$, depending on the target distribution. In comparison, the cost of the traditional MCMC methods such as RWM or MALA is of order $\epsilon^{-2}$, at least when the dimensionality of the small component is more than $1$. Therefore, there is a robustness advantage to using PDMPs in this context.

翻译：分段确定性马尔可夫过程（PDMPs）是一类结合确定性流与跳跃的连续时间马尔可夫过程。近年来，PDMPs作为传统马尔可夫链蒙特卡洛（MCMC）方法的潜在替代方案，引起了蒙特卡洛学界的广泛关注。常用的PDMP方法实例——Zig-Zag采样器和弹跳粒子采样器——已展现出令人瞩目的理论特性，但关于其对目标密度极端依赖性或各向同性的鲁棒性，目前仍知之甚少。研究发现，各向异性可能导致PDMPs混合效率低下，本文将在具有典型意义的重要高斯案例中详细探讨这一效应。为此，本文采用多尺度分析框架。研究结果表明：当高斯目标分布具有$1$阶和$\epsilon$阶两种尺度时，弹跳粒子采样器的计算成本为$\epsilon^{-1}$阶，而Zig-Zag采样器的计算成本则依据目标分布特性，处于$\epsilon^{-1}$或$\epsilon^{-2}$阶。相比之下，传统MCMC方法（如随机游走梅特罗波利斯算法RWM或马尔可夫自适应朗之万算法MALA）的计算成本通常为$\epsilon^{-2}$阶（至少当小尺度分量的维度超过$1$时）。因此，在此情境下，使用PDMPs具有鲁棒性优势。