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We study two important aspects of the diffusion of a free particle in the presence of a timedependent control parameter. The latter is represented by a friction coefficient that is a given function of time. We solve the stochastic Liouville equation (the Fokker-Planck equation) for the probability density of the particle in phase space, i. e., in both position and velocity. The exact solution is then used to analyze the behavior of (i) the variance in the position, a global characterizer of the system; and (ii) the mean rate of crossings of an arbitrary threshold in the position, a local characterizer. The former is the more conventional descriptor of diffusive processes, but the latter provides valuable complementary information on the dynamical behavior. Depending on the long-time behavior of the friction coefficient, the asymptotic behaviors of both these characterizers vary, and exhibit several cross-overs. This helps elucidate the nature of the interplay between the destabilizing effects of the noise and the stabilizing tendency of the damping, as the latter undergoes a controlled variation in time.

Threshold-crossing statistics in diffusion with a time-dependent control parameter