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

TL;DR

This paper characterizes regular diffusions on star graphs as time-changed Walsh Brownian motions, providing new representations, occupation time formulas, and conditions for Feller properties, extending classical diffusion results to complex graph structures.

Contribution

It introduces a novel representation of diffusions on star graphs as time-changed Walsh Brownian motions and extends classical diffusion theory to these graph structures.

Findings

Representation of diffusions as time-changed Walsh Brownian motions

Occupation times formula analogous to Itô--McKean formula

Conditions for Feller and Feller--Dynkin properties

Abstract

We establish the representation of general regular diffusions on star-shaped graphs as time-changed Walsh Brownian motions. These are regular continuous Markov processes described locally by a family generalized second order differential operators defined on every edge and a gluing condition at the junction vertex. This allows us to prove two additional results: (i) A representation of diffusions with sticky gluing conditions as time-changes of diffusions governed by the same differential operators but with non-sticky gluing conditions. (ii) An occupation times formula for such diffusions, analogous to the classical It\^o--McKean formula for one-dimensional diffusions. Additionally, we prove two results of independent interest. First, conditions under which a diffusion on the star graph is Feller and Feller--Dynkin, extending classical results for one-dimensional diffusions. Second, the existence uniqueness of solutions to the Dirichlet problem on the unit disk of the star graph for a general diffusion operator and explicit expressions for its solution.