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 anisotropy 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 ɛ, the computational cost of the Bouncy Particle Sampler is of order ɛ⁻¹, and the computational cost of the Zig-Zag sampler is ɛ⁻² . In comparison, the cost of the traditional MCMC methods such as RWM is of order ɛ⁻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.

Piecewise deterministic Markov processes are an important new tool in the design of Markov chain Monte Carlo algorithms. Two examples of fundamental importance are the bouncy particle sampler (BPS) and the zig–zag process (ZZ). In this paper scaling limits for both algorithms are determined. Here the dimensionality of the space tends towards infinity and the target distribution is the multivariate standard normal distribution. For several quantities of interest (angular momentum, first coordinate and negative log-density) the scaling limits show qualitatively very different and rich behaviour. Based on these scaling limits the performance of the two algorithms in high dimensions can be compared. Although for angular momentum both processes require only a computational effort of O(d) to obtain approximately independent samples, the computational effort for negative log-density and first coordinate differ: for these BPS requires O(d²) computational effort whereas ZZ requires O(d). Finally we provide a criterion for the choice of the refreshment rate of BPS.