Brownian Motions on Metric Graphs

TL;DR

This paper characterizes Brownian motions on metric graphs through their generators as Laplace operators with Wentzell boundary conditions, and constructs such processes given boundary conditions, bridging stochastic processes and boundary value problems.

Contribution

It provides a complete characterization of Brownian motions on metric graphs via boundary conditions and constructs these processes from specified boundary data.

Findings

Generators are Laplace operators with Wentzell boundary conditions.

Constructs Brownian motions from given boundary conditions.

Establishes a correspondence between boundary conditions and stochastic processes.

Abstract

Brownian motions on a metric graph are defined. Their generators are characterized as Laplace operators subject to Wentzell boundary at every vertex. Conversely, given a set of Wentzell boundary conditions at the vertices of a metric graph, a Brownian motion is constructed pathwise on this graph so that its generator satisfies the given boundary conditions.