Spectral analysis of the zigzag process

Journal article (2022)

Authors

G.N.J.C. Bierkens Statistics -

Sjoerd M. Verduyn Lunel Universiteit Utrecht

Research Group

Statistics () (TU Delft)

DOI: https://doi.org/10.1214/21-AIHP1188

Perturbation theory Exponential ergodicity Piecewise deterministic Markov process Markov semigroup Non-reversible Markov process Spectral theory Telegraph process Zigzag process

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Published Date

2022

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Statistics

Abstract

The zigzag process is a variant of the telegraph process with position dependent switching intensities. A characterization of the L2-spectrum for the generator of the one-dimensional zigzag process is obtained in the case where the marginal stationary distribution on R is unimodal and the refreshment intensity is zero. Sufficient conditions are obtained for a spectral mapping theorem, mapping the spectrum of the generator to the spectrum of the corresponding Markov semigroup. Furthermore results are obtained for symmetric stationary distributions and for perturbations of the spectrum, in particular for the case of a non-zero refreshment intensity. In the examples we consider (including a Gaussian target distribution) a slight increase of the refreshment intensity above zero results in a larger L2-spectral gap, corresponding to an improved convergence in L2.

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