The Annals of Probability

Diffusion processes in thin tubes and their limits on graphs

Sergio Albeverio and Seiichiro Kusuoka

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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.

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

First available in Project Euclid: 8 October 2012

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Primary: 60J60: Diffusion processes [See also 58J65] 60H30: Applications of stochastic analysis (to PDE, etc.) 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60]
Secondary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 35K15: Initial value problems for second-order parabolic equations 34B45: Boundary value problems on graphs and networks

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


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.

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