Escape over a fluctuating barrier: Limits of small and large correlation times

We investigate the problem of diffusion across a randomly fluctuating barrier in the presence of thermal noise. The barrier fluctuations an induced by an Ornstein-Uhlenbeck noise the strength Q of which is assumed to depend on the noise correlation time tau. In the vicinity of the limits of zero and infinite tau we calculate the exact formulas for the first two terms of the expansion in powers of tau of the mean first-passage time (MFPT) over the top of the barrier. The results are strongly conditioned by the form of the tau dependence of Q. The main conclusion is that the nonmonotonic tau dependence of the MFPT is generic, while the monotonicity of the MFPT occurs only in some specific cases. When tau increases from zero, for a class of barrier noises with Q increasing faster than linearly one should observe ''resonant activation,'' i.e., a minimum of the MFPT as a function of tau. The appearance of a maximum, called ''inhibition of activation,'' is also possible provided that the noise variance D increases faster than linearly as a function of 1/tau in the vicinity of the limit 1/tau-->0. Both kinds of extrema may also appear simultaneously. These effects depend neither on the shape of the barrier nor on its disturbance. If Q(tau) [or D(1/tau)] varies Linearly or slower as tau (1/tau) increases from zero, then the peculiarities of the perturbed barrier become essential and any type of tau dependence of the MFPT, also a monotonic one, is possible. The specific analogy between the properties of the MFPT for tau-->0 and for tau-->infinity is stressed.