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Historically time-reversibility of the transitions or processes underpinning Markov chain Monte Carlo methods (MCMC) has played a key r\^ole in their development, while the self-adjointness of associated operators together with the use of classical functional analysis techniques on Hilbert spaces have led to powerful and practically successful tools to characterize and compare their performance.
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The Zig-Zag process and super-efficient sampling for bayesian analysis of big data
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Spectral analysis of the Zig-Zag process
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