Spectral analysis of the Zig-Zag process on the torus

In this thesis, we analyse the spectrum of the generator of the one-dimensional Zig-Zag process defined on the torus $\mathbb{T}$. This is a piecewise deterministic Markov process (PDMP) used in Monte Carlo Markov chain methods (MCMC) for sampling from a probability distribution and calculating integrals \cite{Rejectionfree}, \cite{ZigZag}, ...

Duality in interacting particle systems out of equilibrium

In this thesis we study a class of interacting particle systems sharing a duality property. This class includes the Symmetric Inclusion Process (SIP(2k)), the Symmetric Exclusion Process (SEP(2j)) and the Independent Random Walkers (IRW). When these systems are in equilibrium (namely they are isolated from the exterior) they admit stationary...