Quasi-processes for branching Markov chains

Potential theory is a central tool to understand and analyse Markov processes. In this article, we develop its probabilistic counterpart for branching Markov chains. More explicitly, we consider analogues of quasi-processes that involve branching (branching quasi-processes). Branching quasi-processes can be characterized in terms of their occupation measures. In the case where a particular decorability condition is satisfied, there is an isomorphism between the cone of branching quasi-processes and the cone of excessive measures, where the respective excessive measures are the occupation measures of the branching quasi-processes. Taking a branching quasi-process as intensity measure for a Poison point process we obtain random interlacements that incorporate branching. In the particular case, where individuals produce offspring with mean less than or equal to one, we provide an explicit construction based on classical interlacements. We make strong use of the particular additive structure of branching Markov chains and a main technical tool will be a spinal representation for the involved branching processes.