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}, \cite{Bouncy}. We show for Lipschitz potentials $U$ and bounded refreshment rates $\lambda_0 \in L^{\infty}(\mathbb{T})$ that the spectral gap $\kappa = \sup\{\operatorname{Re} \lambda : \lambda \in \sigma(\mathcal{L})\} \setminus \{0 \}$ of the associated $J$-self-adjoint generator $\mathcal{L}$ on $L^2(\mathbb{T} ,\nu)$ and $C(\mathbb{T} \times \{+1,-1\})$ is positive. Moreover, we give two lower bounds for $\kappa$ by making use of one of the Schur complements associated with a block operator that is unitarily equivalent to $\mathcal{L}$. In addition we show that the spectrum of $L^2(\mathbb{T} ,\nu)$ and $C(\mathbb{T} \times \{+1,-1\})$ are the same and that the generator defined on both spaces generates a contraction semigroup. Under the assumption of unimodality of the potential $U$ and a zero refreshment rate, we show that a vertical "asymptotic line" exists to which all of the eigenvalues converge. Furthermore, we show that a spectral mapping theorem exists where, due to the spectral line, the spectrum of the semigroup can become uncountable or countable depending on the time parameter of the semigroup $P(t)$ generated by $\mathcal{L}$. Lastly, we show that a discretisation of the spectrum generates a semigroup that converges uniformly on each bounded time interval to the semigroup of the Zig-Zag process and we use these discretisations to numerically analyse the behaviour of general potentials and refreshment rates.

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 measures that are also reversible product space-homogeneous measures. However when the systems are in contact with reservoirs that keep them out of equilibrium, the reversibility is lost and the stationary microscopic distributions are unknown with the exception for the SEP(1) (the Symmetric Exclusion Process) and the IRW. In order to examine the non-equilibrium stationary distributions for the whole class of processes, we make use of duality between these processes and particle systems with absorbing boundaries. Our first main results are in section 6, where we give explicit formulas for the absorption probabilities for the dual systems with 1 and 2 particles. Then we use this result to compute an explicit formula for the variance and covariance of the sites occupation numbers for the many-particles systems with density reservoirs in the non-equilibrium stationary distribution in section 7. Then we identify three possible scaling regimes for the density field: a deterministic regime where the variance vanishes and particles are expected to converge to independent Brownian motions; a sticky regime where the variance is finite and particles are expected to converge to sticky Brownian motions; and finally an absorbing regime where the variance is infinite and particles are expected to converge to coalescing Brownian motions. In the last part of the thesis (section 8) we start the analysis of the dynamics of the particle systems.