General diffusions on the star graph as time-changed Walsh Brownian motion

Abstract: We establish a representation of general regular diffusions on star-shaped graphs as time-changed Walsh Brownian motions. These are Markov processes with continuous sample paths whose law on each edge is described locally by generalized second-order operators, a gluing condition at the junction vertex, and boundary conditions. The representation is built upon two key results: (i) a time-change representation for the distance-to-origin process, and (ii) a probabilistic interpretation of the gluing condition. This result is leveraged to derive an occupation times formula for these processes. Additionally, we prove two results of independent interest. First, we provide conditions under which a diffusion on the star graph is Feller and Feller--Dynkin, extending classical results for one-dimensional diffusions. Second, we establish the existence and uniqueness of solutions to the Dirichlet problem on the unit disk of the star graph, along with an explicit expression for the corresponding Green function.