Expansion of the Master Equation for One-Dimensional Random Walks with Boundary

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Abstract
In order to understand the behavior of coarse‐grained equations in the presence of a boundary, the following model is investigated. A homogeneous one‐dimensional random walk is bounded on one side by some boundary conditions of rather arbitrary form. The corresponding master equation is approximated by the Fokker‐Planck equation plus partial differential equations for the higher orders. The boundary condition for the Fokker‐Planck approximation is well known; but those for the higher order terms are here derived. To the second order they amount to a virtual displacement of the boundary. The case of a two‐step random walk, however, gives rise to an unexpected complication, inasmuch as nonpropagating solutions of the master equation cannot be ignored in the boundary condition, although they do not contribute to the differential equations themselves.