A drop of ink falls from my pen. . . it comes to earth, I know not when

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Abstract
The authors obtain, for a Brownian particle in a uniform force field, the mean and asymptotic first-passage times as functions of the particle's initial position and velocity, with the recurrence times given as a special case. They discuss the region of phase space for which the diffusion model of Brownian motion provides an adequate approximation, and conclude that there is no possibility of obtaining the recurrence times within that model. They find that the nature of the boundary-value problem is profoundly altered when the motion is treated as a process in phase rather than configuration space, because the time-development operator is then parabolic rather than elliptic. They argue that such a change in the treatment of Brownian motion places it within the sphere of transport theory rather than diffusion theory, and that, consequently, results such as ours have relevance to the study of phenomena such as radiative transfer and neutron transport.

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