Sticky Brownian motions on star graphs

Abstract

This paper is concerned with the construction of Brownian motions and related stochastic processes in a star graph, which is a non-Euclidean structure where some features of the classical modeling 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 the 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 \(\varPhi \). 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 conditions in the vertex that can be written in terms of a Caputo-Džrbašjan fractional derivative defined by the singular measure \(\varPhi \). Extensions to general graph structures can be given by applying to our results a localisation technique.