Boundary behavior of diffusion approximations to Markov jump processes

C. Knessl*, B. J. Matkowsky, Z. Schuss, C. Tier

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We show that diffusion approximations, including modified diffusion approximations, can be problematic since the proper choice of local boundary conditions (if any exist) is not obvious. For a class of Markov processes in one dimension, we show that to leading order it is proper to use a diffusion (Fokker-Planck) approximation to compute mean exit times with a simple absorbing boundary condition. However, this is only true for the leading term in the asymptotic expansion of the mean exit time. Higher order correction terms do not, in general, satisfy simple absorbing boundary conditions. In addition, the diffusion approximation for the calculation of mean exit times is shown to break down as the initial point approaches the boundary, and leads to an increasing relative error. By introducing a boundary layer, we show how to correct the diffusion approximation to obtain a uniform approximation of the mean exit time. We illustrate these considerations with a number of examples, including a jump process which leads to Kramers' diffusion model. This example represents an extension to a multivariate process.

Original languageEnglish
Pages (from-to)245-266
Number of pages22
JournalJournal of Statistical Physics
Issue number1-2
StatePublished - Oct 1986


  • Jump process
  • boundary conditions
  • diffusion approximations
  • master equation
  • singular perturbations


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