Interacting Jump Processes Preserve Semi-Global Markov Random Fields on Path Space

Consider a system of interacting particles indexed by the nodes of a graph whose vertices are equipped with marks representing parameters of the model such as the environment or initial data. Each particle takes values in a countable state space and evolves according to a (possibly non-Markovian) continuous-time pure jump process whose jump intensities depend only on its own state (or history) and marks as well as the states (or histories) and marks of particles and edges in its neighborhood in the graph. Under mild conditions on the jump intensities, it is shown that the trajectories of the interacting particle system exhibit a certain local or semi-global Markov random field property whenever the initial condition satisfies the same property. Our results complement recent works that establish the preservation of a local second-order Markov random field property for interacting diffusions. Our proof methodology in the context of jump processes is different, and works directly on infinite graphs, thereby bypassing any limiting arguments. Our results apply to models arising in diverse fields including statistical physics, neuroscience, epidemiology and opinion dynamics, and have direct applications to the study of marginal distributions of interacting particle systems on Cayley trees.