Journal of Statistical Physics

, Volume 158, Issue 1, pp 192–230 | Cite as

Exit Time Distribution in Spherically Symmetric Two-Dimensional Domains

  • J.-F. Rupprecht
  • O. Bénichou
  • D. S. Grebenkov
  • R. Voituriez


The distribution of exit times is computed for a Brownian particle in spherically symmetric two-dimensional domains (disks, angular sectors, annuli) and in rectangles that contain an exit on their boundary. The governing partial differential equation of Helmholtz type with mixed Dirichlet–Neumann boundary conditions is solved analytically. We propose both an exact solution relying on a matrix inversion, and an approximate explicit solution. The approximate solution is shown to be exact for an exit of vanishing size and to be accurate even for large exits. For angular sectors, we also derive exact explicit formulas for the moments of the exit time. For annuli and rectangles, the approximate expression of the mean exit time is shown to be very accurate even for large exits. The analysis is also extended to biased diffusion. Since the Helmholtz equation with mixed boundary conditions is encountered in microfluidics, heat propagation, quantum billiards, and acoutics, the developed method can find numerous applications beyond exit processes.


Exit time Residence time Mixed boundary condition Helmholtz equation Active transport Microfluidic Heat transfer 



O.B. is supported by the ERC Starting Grant No. FPTOpt-277998. D.G. is supported by an ANR project “INADILIC”.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • J.-F. Rupprecht
    • 1
  • O. Bénichou
    • 1
  • D. S. Grebenkov
    • 2
  • R. Voituriez
    • 1
  1. 1.Sorbonne Universités, UPMC Univ Paris 06, UMR 7600Laboratoire de Physique Théorique de la Matière Condensée75005 ParisFrance
  2. 2.Laboratoire de Physique de la Matière Condensée (UMR 7643)CNRS – Ecole PolytechniquePalaiseau CedexFrance

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