Sticky Brownian motions on star graphs

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.