Critical Slowing Down in Stochastic Precesses. II ---Noise-Induced Long-Time Tail in Random Growing-Rate Models---
Exact solutions of the random growing-rate models dx/dt = (γ + η (t))x- gxm for Gaussian white noise η(t) are obtained for an arbitrary value of m(>1) to give the noise-induced long-time tail < xp(t)> ≃ t-1/2 for large t at the critical point for any positive value of p. This confirms the phenomenological arguments in the first paper of this series. That is, this noise-induced long-time tail appears due to the balance (or cancellation) between the nonlinear term and noise term. Another intuitive explanation of this noise-induced long-time tail is presented on the basis of the asymptotic behavior of the normalization factor for the distribution function. A slowing down of <xp(t)> for p<0 at a `transition point' γ =γ (p) = |p| ∊ is also discussed.