Analytic and stochastic description of Brownian motions on star graphs

TL;DR

This paper characterizes all Feller processes on finite star graphs, extending classical Brownian motion theory to complex graph structures with a focus on stochastic and analytic descriptions.

Contribution

It provides a comprehensive classification of Feller processes on star graphs, bridging stochastic processes with multi-armed bandit theory.

Findings

Complete description of Feller processes on star graphs

Extension of Brownian motion theory to graph structures

Connection to multi-armed bandit framework

Abstract

We provide a detailed description of all possible Feller processes on infinite} star graphs with finite number of edges, processes that while away from the graph's center behave like a one-dimensional Brownian motion. The description can be seen as a continuation of the seminal paper by It\^o and McKean (devoted to Brownian motions on the half-line), recast from the perspective of the theory of multi-armed bandits.