Exact solution of a nonlinear Langevin equation with applications to photoelectron counting and noise-induced instability

A stochastic differential equation of the Langevin type involving a linear and a quadratic noise is considered and an exact solution for the stochastic average of this equation is obtained. This solution is employed to calculate the effect of amplitude fluctuations of an electromagnetic field on the photoelectron counting, and an explicit expression is obtained for the photon counting probability. Finally, the steady state limit is studied and the possibility of an instability induced by the noise is discussed.