Brownian Motions on Metric Graphs

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.

43 pages, 7 figures. 2nd revision of our article 1102.4937: The introduction has been modified, several references were added. This article will appear in the special issue of Journal of Mathematical Physics celebrating Elliott Lieb's 80th birthday

References in corpus (5)

• On the constructions of the skew Brownian motion
• The inverse scattering problem for metric graphs and the traveling salesman problem
• Exit and Occupation times for Brownian Motion on Graphs with General Drift and Diffusion Constant
• Brownian Motions on Metric Graphs: Feller Brownian Motions on Intervals Revisited
• Construction of the Paths of Brownian Motions on Star Graphs

Cited by in corpus (13)

• Non-self-adjoint graphs
• Schrödinger and polyharmonic operators on infinite graphs: Parabolic well-posedness and p-independence of spectra
• Maximal-quasi-accretive Laplacians on finite metric graphs
• Tropical Gaussians: A Brief Survey
• Hidden symmetries in non-self-adjoint graphs
• Construction of the Paths of Brownian Motions on Star Graphs
• Reaction-diffusion on metric graphs and conversion probability
• Impediments to diffusion in quantum graphs: geometry-based upper bounds on the spectral gap
• Constructing exchangeable pairs by diffusion on manifolds and its application
• Brownian Motions on Star Graphs with Non-Local Boundary Conditions
• Brownian Motions on Metric Graphs with Non-Local Boundary Conditions I: Characterization
• Brownian Motions on Metric Graphs with Non-Local Boundary Conditions II: Construction
• Martin boundary of Brownian motion on Gromov hyperbolic metric graphs