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Spectral analysis of the zigzag process. (English. French summary) Zbl 1519.37005

The zigzag process is a kinetic piecewise deterministic Markov process, with constant speed, designed to sample a given Gibbs measure in Markov chain Monte Carlo methods. Its efficiency relies on its ergodicity properties, linked to the long-time convergence of its associated semi-group in \(L^2\). This is in turns related to the spectral properties of its generator, which is the topic of this work. On the one hand, a technical obstacle is that the generator is not self-adjoint, local and compact. On the other hand, the two main ingredients that make the investigation possible are: first, the fact that the generator is conjugated with its adjoint by a flip of velocity and, second, the restriction to the one-dimensional unimodal setting with no refreshment, so that computing resolvent or eigenfunctions can be obtained explicitly by solving a system of two one-dimensional ordinary differential equations. The case of a non-zero refreshment rate is a matter of perturbation analysis (as the refreshment operator is bounded), i.e., the linearization of the spectrum with respect to a small refreshment rate is given. The first-order effect of a small refreshment rate can then be computed numerically. This is done on examples, e.g., exponential tails and Gaussian tails. It is observed that, with respect to the case with no refreshment, a small refreshment increases the spectral gap.

MSC:

37A50 Dynamical systems and their relations with probability theory and stochastic processes
37A30 Ergodic theorems, spectral theory, Markov operators
37M25 Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.)
47A10 Spectrum, resolvent
60J25 Continuous-time Markov processes on general state spaces
60H25 Random operators and equations (aspects of stochastic analysis)
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