Brownian Motions on Metric Graphs with Non-Local Boundary Conditions II: Construction

TL;DR

This paper presents a comprehensive pathwise construction method for discontinuous Brownian motions on metric graphs with non-local boundary conditions, enabling detailed analysis of such stochastic processes.

Contribution

It introduces a novel local decomposition and pasting technique to construct Brownian motions with non-local jumps on metric graphs, expanding existing theoretical frameworks.

Findings

Constructed Brownian motions with non-local jumps on metric graphs.

Verified the generator of the constructed processes.

Provided a systematic method for handling non-local boundary conditions.

Abstract

A pathwise construction of discontinuous Brownian motions on metric graphs is given for every possible set of non-local Feller-Wentzell boundary conditions. This construction is achieved by locally decomposing the metric graphs into star graphs, establishing local solutions on these partial graphs, pasting the solutions together, introducing non-local jumps, and verifying the generator of the resulting process.