Ergodicity of the zigzag process

Abstract

The zigzag process is a variant of the telegraph process with position dependent switching intensities. A characterization of the L 2spectrum 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.Keywords and phrases: spectral theory, non-reversible Markov process, Markov semigroup, zigzag process, exponential ergodicity, perturbation theory. * This work is part of the research programme 'Zig-zagging through computational barriers' with project number 016.Vidi.189.043, which is financed by the Netherlands Organisation for Scientific Research (NWO).

The Zig-Zag process and super-efficient sampling for Bayesian analysis of big data

“…Based on numerous experiments, we conjecture that the canonical multi-dimensional Zig-Zag process is ergodic in general under only mild conditions. A detailed investigation of ergodicity will be the subject of a forthcoming paper (Bierkens, Roberts and Zitt, 2017).…”

“…Since the first version of this paper was conceived already several other related theoretical and methodological papers have appeared. In particular we mention here results on exponential ergodicity of the BPS (Deligiannidis, Bouchard-Côté and Doucet, 2017) and ergodicity of the multi-dimensional Zig-Zag process (Bierkens, Roberts and Zitt, 2017). The Zig-Zag process has the advantage that it is ergodic under very mild conditions, which in particular means that we are not required to choose a refreshment rate.…”

Constructing Sampling Schemes via Coupling: Markov Semigroups and Optimal Transport

“…For ease of exposition, we furthermore restrict our attention to the one-dimensional case. The treatment here follows [10] and [11] in style and notation.…”

“…See[9] and[11]. Let us mention in particular that ergodicity is guaranteed whenever the excess switching rate γ is strictly positive.…”

“…For β ∈ [0, π 4 ], the function 2 sin β cos is monotone, taking values in [0, 1]. We chose this parametrisation in order for it to be consistent with(52) 11. However, often one aims to approximate the solution to the Poisson equation in order to use it as a control variate, see for instance[26,62,76].…”