The Analogue of Bohm-Bell Processes on a Graph

Bohm-Bell processes, of interest in the foundations of quantum field theory, form a class of Markov processes $Q_t$ generalizing in a natural way both Bohm's dynamical system in configuration space for nonrelativistic quantum mechanics and Bell's jump process for lattice quantum field theories. They are such that at any time $t$ the distribution of $Q_t$ is $|\psi_t|^2$ with $\psi$ the wave function of quantum theory. We extend this class here by introducing the analogous Markov process for quantum mechanics on a graph (also called a network, i.e., a space consisting of line segments glued together at their ends). It is a piecewise deterministic process whose innovations occur only when it passes through a vertex.