Brownian Motions on Star Graphs with Non-Local Boundary Conditions

TL;DR

This paper characterizes Brownian motions on star graphs with complex boundary behaviors, including jumps and stickiness, by identifying their generators and providing explicit pathwise constructions for various boundary conditions.

Contribution

It introduces a comprehensive framework for Brownian motions on star graphs with non-local boundary conditions, extending existing constructions to include jumps and stickiness.

Findings

Generators are Laplace operators with non-local Feller-Wentzell boundary conditions.

Pathwise descriptions are constructed for finite and infinite jump measures.

These processes serve as building blocks for Brownian motions on general metric graphs.

Abstract

Brownian motions on star graphs in the sense of It\^o-McKean, that is, Walsh processes admitting a generalized boundary behavior including stickiness and jumps and having an angular distribution with finite support, are examined. Their generators are identified as Laplace operators on the graph subject to non-local Feller-Wentzell boundary conditions. A pathwise description is achieved for every admissible boundary condition: For finite jump measures, a construction of Kostrykin, Potthoff and Schrader in the continuous setting is expanded via a technique of successive killings and revivals; for infinite jump measures, the pathwise solution of It\^o-McKean for the half line is analyzed and extended to the star graph. These processes can then be used as main building blocks for Brownian motions on general metric graphs with non-local boundary conditions.