  This paper is concerned with the construction of several stochastic processes
in a star graph, that is a non-euclidean structure where some features of the
classical modelling fail. We propose a model for trapping phenomena with
characterization of the traps in terms of a singular measure. This measure also
defines a non-local operator by means of which we introduce a non-local dynamic
condition for the parabolic problem on the star graph. We study semi-Markov
processes on the rays of the graph in order to obtain a probabilistic
representation of the motion on the whole graph. Extensions to general graph
structures can be given by applying our results on star graphs.
