Brownian motion with an absorbing boundary

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Abstract
We consider the problem of determining the joint position–velocity distribution function for a Brownian particle confined by an absorbing boundary. The approach used here, which we believe to be novel in this context, involves an integral equation formulation in terms of the distribution function, for the unrestricted problem, of first turning points in the half‐space that does not contain the initial source. The advantage of this formulation is that it leads to an equation of the Wiener–Hopf type, so that in principle explicit solutions can be found by standard methods. An approximate solution is found for the case of large friction which reproduces the general features of the solution found in the theory based on the use of the diffusion equation; this result illustrates the potential of our formulation with regard to obtaining an exact solution to the absorbing boundary problem.