Diffusion driven by a constant field, and opposed by a velocity‐dependent diffusion coefficient that decreases to zero at large velocity, leads to the phenomenon of ‘‘runaway.’’ It is studied here in the case of a one‐dimensional velocity space, when the Fokker–Planck equation is equivalent to an interesting Schro¨dinger equation. The runaway current is extracted from the smallest eigenvalue, after the equation has been solved by the method of matched asymptotic expansions. There is discussion of connections between our approach, the conventional approach, and the classical two‐dimensional results.