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}, \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.
Applied Mathematics