Diffusion processes in thin tubes and their limits on graphs

Sergio Albeverio, Seiichiro Kusuoka

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)2131-2167
Number of pages37
JournalAnnals of Probability
Volume40
Issue number5
DOIs
Publication statusPublished - 2012 Jan 1

Keywords

  • Diffusion processes
  • Dirichlet boundary conditions
  • Kirchhoff boundary conditions
  • Neumann boundary conditions
  • Processes on graphs
  • Thin tubes
  • Weak convergence

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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