  Potential theory is a central tool to understand and analyse Markov
processes. In this article, we develop its probabilistic counterpart for
branching Markov chains. Specifically, we examine versions of quasi-processes
or interlacements that incorporate branching, referred to as branching
quasi-processes. These processes are characterized by their occupation
measures. If a certain decorability condition is fulfilled, there's an
isomorphism between the set of branching quasi-processes and the set of
excessive measures, where the excessive measures correspond to the occupation
measures of the branching quasi-processes. Utilizing a branching quasi-process
as an intensity measure for a Poisson point process leads to the formulation of
random interlacements with branching. In cases where individuals reproduce with
an average rate of one or less, we detail a construction that draws on
classical interlacements. Our approach significantly employs the additive
structure of branching Markov chains, with the spinal representation of the
branching processes serving as a crucial technical tool.
