Open Access
Translator Disclaimer
September 2012 Diffusion processes in thin tubes and their limits on graphs
Sergio Albeverio, Seiichiro Kusuoka
Ann. Probab. 40(5): 2131-2167 (September 2012). DOI: 10.1214/11-AOP667


The present paper is concerned with diffusion processes running on tubular domains with conditions on nonreaching the boundary, respectively, reflecting at the boundary, and corresponding processes in the limit where the thin tubular domains are shrinking to graphs. The methods we use are probabilistic ones. For shrinking, we use big potentials, respectively, reflection on the boundary of tubes. We show that there exists a unique limit process, and we characterize the limit process by a second-order differential generator acting on functions defined on the limit graph, with Kirchhoff boundary conditions at the vertices.


Download Citation

Sergio Albeverio. Seiichiro Kusuoka. "Diffusion processes in thin tubes and their limits on graphs." Ann. Probab. 40 (5) 2131 - 2167, September 2012.


Published: September 2012
First available in Project Euclid: 8 October 2012

zbMATH: 1267.60090
MathSciNet: MR3025713
Digital Object Identifier: 10.1214/11-AOP667

Primary: 58J65 , 60H30 , 60J60
Secondary: 34B45 , 35K15 , 60J35

Keywords: Diffusion processes , Dirichlet boundary conditions , Kirchhoff boundary conditions , Neumann boundary conditions , processes on graphs , thin tubes , weak convergence

Rights: Copyright © 2012 Institute of Mathematical Statistics


Vol.40 • No. 5 • September 2012
Back to Top