Sticky Brownian motions on star graphs

This paper is concerned with the construction of Brownian motions and related stochastic processes in a star graph, that is a non-euclidean structure where some features of the classical modelling fail. We propose a probabilistic construction of the Sticky Brownian motion by slowing down the Brownian motion when in the vertex of the star graph. Later, we apply a random change of time to previous construction, which leads to a trapping phenomenon in the vertex of the star graph, with characterization of the trap in terms of a singular measure $\Phi$. The process associated to this time change is described here and, moreover, we show that it defines a probabilistic representation of the solution to a heat equation type problem on the star graph with non-local dynamic condition in the vertex that are described in terms of a Caputo-Djrbashian fractional derivative defined by the singular measure $\Phi$. Extensions to general graph structures can be given by applying to our results a localisation technique.