Brownian Motions on Metric Graphs with Non-Local Boundary Conditions I: Characterization

TL;DR

This paper classifies Brownian motions on metric graphs with complex boundary behaviors, establishing their Feller property and characterizing their generators through non-local boundary conditions, especially on star graphs.

Contribution

It provides a comprehensive classification and characterization of Brownian motions on metric graphs with non-local boundary conditions, including a complete generator description for star graphs.

Findings

Established the Feller property for these processes.

Identified non-local Feller-Wentzell boundary conditions.

Provided a complete generator description for star graphs.

Abstract

A classification for Brownian motions on metric graphs, that is, right continuous strong Markov processes which behave like a one-dimensional Brownian motion on the edges and feature effects like Walsh skewness, stickiness and jumps at the vertices, is obtained. The Feller property of these processes is proved, and the boundary conditions of their generators are identified as non-local Feller-Wentzell boundary conditions. By using a technique of successive revivals, a complete description of the generator is achieved for Brownian motions on star graphs.