Uniform asymptotic expansions in dynamical systems driven by colored noise

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Abstract
We construct asymptotic expansions of the (quasi)stationary probability density function and of the mean first-passage time over a potential barrier, for bistable systems driven by weak wideband Gaussian colored noise, when the intensity ε and the correlation time τ of the noise are both small. Previous analyses have led to a variety of often contradictory results and to considerable confusion, which stem from the fact that the problem depends on two small parameters. This results in different expansions, with different ranges of validity, depending on the relative magnitudes of ε and τ. In contrast, we derive expansions that are uniformly valid throughout the entire parameter range of interest. In addition, we identify restrictions on the ranges of validity, in terms of the total power output ε/τ of the noise, of previously derived expansions. We show that only if the power output ε/τ becomes infinite can previously derived expansions be valid. Our results, when specialized to this case, reduce to expansions previously derived. Outside the restricted range, i.e., for finite or vanishingly small power outputs, our expansions contain terms which are new, and which may in fact dominate previously computed terms. In contrast to the use of one-dimensional diffusional approximations previously employed, our approach is based on the exact two-dimensional Fokker-Planck equation. Singular perturbation techniques, previously developed by the authors, are employed to systematically derive the asymptotic expansions.