## The Annals of Probability

### Diffusion processes in thin tubes and their limits on graphs

#### 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.

#### Article information

Source
Ann. Probab., Volume 40, Number 5 (2012), 2131-2167.

Dates
First available in Project Euclid: 8 October 2012

https://projecteuclid.org/euclid.aop/1349703318

Digital Object Identifier
doi:10.1214/11-AOP667

Mathematical Reviews number (MathSciNet)
MR3025713

Zentralblatt MATH identifier
1267.60090

#### Citation

Albeverio, Sergio; Kusuoka, Seiichiro. Diffusion processes in thin tubes and their limits on graphs. Ann. Probab. 40 (2012), no. 5, 2131--2167. doi:10.1214/11-AOP667. https://projecteuclid.org/euclid.aop/1349703318

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