📝 SummarXiv

Abstract

We provide a general model for Brownian motions on metric graphs with interactions. In a general setting, for (sticky) Brownian propagations on edges, our model provides a characterization of lifetimes and holding times on vertices in terms of (jumping) Brownian accumulation of energy associated with that vertices. Propagation and accumulation are given by drifted Brownian motions subjected to non-local (also dynamic) boundary conditions. As the continuous (sticky) process approaches a vertex, then the right-continuous process has a restart (resetting), it jumps randomly away from the zero-level of energy. According with this new energy, the continuous process can start (or not) as a new process in a randomly chosen edge. We provide a self-contained presentation with a detailed construction of the model. The model well extends to a higher order of interactions, here we provide a simple case and focus on the analysis of earthquakes. Earthquakes are notoriously difficult to study. They build up over long periods and release energy in seconds. Our goal is to introduce a new model, useful in many contexts and in particular in the difficult attempt to manage seismic risks.