Brownian Motions on Metric Graphs I - Definition, Feller Property, and Generators

TL;DR

This paper defines Brownian motions on metric graphs, proves their Feller property, characterizes their generators, and extends Feller's theorem to this setting, advancing the mathematical understanding of stochastic processes on complex structures.

Contribution

It introduces a rigorous definition of Brownian motions on metric graphs and characterizes their generators, extending classical results to this new context.

Findings

Brownian motions on metric graphs are well-defined.

The Feller property is established for these processes.

A version of Feller's theorem for metric graphs is proved.

Abstract

Brownian motions on a metric graph are defined, their Feller property is proved, and their generators are characterized. This yields a version of Feller's theorem for metric graphs.