DIFFUSION PROCESSES IN THIN TUBES AND THEIR LIMITS ON GRAPHS

• 成果类型: Article
• 署名作者: Albeverio, Sergio; Kusuoka, Seiichiro
• 署名单位: University of Bonn; Kyoto University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/11-AOP667
• 发表日期: 2012
• 页码: 2131-2167
• 关键词: quantum graphs; CONVERGENCE; EQUATIONS; SPECTRA; systems; STATES

摘要

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.