Revisit to the scaling theory of transient phenomena — Generalization to correlated noise and singular perturbation expansion up to infinite order

A scaling theory is formulated in nonlinear Langevin equations with colored noise near the instability point from which the center of the initial distribution is deviated slightly. An intuitive derivation is given as well as perturbational calculations up to the infinite order of the noise. Moments of all orders are calculated in the approximation in which only the most dominant terms are retained in the scaling limit. Hence the scaling distribution function is also obtained. The onset time t0 is derived as a functional of the correlation function of the noise and it is shown to increase when the noise correlation length increases.