In this paper we present a rigorous asymptotic analysis for stochastic
systems with two fast relaxation times. The mathematical model analyzed in this
paper consists of a Langevin equation for the particle motion with
time-dependent force constructed through an infinite dimensional Gaussian noise
process. We study the limit as the particle relaxation time as well as the
correlation time of the noise tend to zero and we obtain the limiting equations
under appropriate assumptions on the Gaussian noise. We show that the limiting
equation depends on the relative magnitude of the two fast time scales of the
system. In particular, we prove that in the case where the two relaxation times
converge to zero at the same rate there is a drift correction, in addition to
the limiting It\^{o} integral, which is not of Stratonovich type. If, on the
other hand, the colored noise is smooth on the scale of particle relaxation
then the drift correction is the standard Stratonovich correction. If the noise
is rough on this scale then there is no drift correction. Strong (i.e.
pathwise) techniques are used for the proof of the convergence theorems.Comment: 35 pages, 0 figures, To appear in SIAM J. MM
