Absorbing-boundary limit for Brownian motion: Demonstration for a model

We consider one-dimensional diffusion with an absorbing boundary in the context of a model that retains the essential ‘‘missing’’ boundary condition feature that complicates the solution of the Fokker-Planck equation for this problem. The model solution is obtained and some of its features are briefly discussed but our primary purpose is to demonstrate a limiting process that transforms the solution of a related diffusion problem to that for the absorbing-boundary problem. This approach, which we refer to as the absorbing-boundary limit, consists of obtaining the solution for the infinite-space case with the physical space -∞<x<∞ characterized by separate friction coefficients for x≶0. The solution obtained will be a function of these two quantities, ${\beta }_1$ for x>0 and ${\beta }_2$ for x<0, and in the limit ${\beta }_2$→0 describes a process for which the Brownian particle diffuses to x=-∞ when it crosses the origin and does not return to x>0. The solution for x>0 is thus identical to that for the case where the origin is an absorbing ‘‘boundary.’’ This limiting process provides a new method for obtaining a solution to the Fokker-Planck equation with an absorbing-boundary condition and may lead to a result that is in closed form and more transparent than the eigenfunction expansions recently obtained by other means.