  We study the effect of additive Brownian noise on an ODE system that has a
stable hyperbolic limit cycle, for initial data that are attracted to the limit
cycle. The analysis is performed in the limit of small noise - that is, we
modulate the noise by a factor $\varepsilon \searrow 0$ - and on a long time
horizon. We prove explicit estimates on the proximity of the noisy trajectory
and the limit cycle up to times $\exp\left(c \varepsilon^{-2}\right)$, $c>0$,
and we show both that on the time scale $\varepsilon^{-2}$ the "'dephasing"
(i.e., the difference between noiseless and noisy system measured in a natural
coordinate system that involves a phase) is close to a Brownian motion with
constant drift, and that on longer time scales the dephasing dynamics is
dominated, to leading order, by the drift. The natural choice of coordinates,
that reduces the dynamics in a neighborhood of the cycle to a rotation, plays a
central role and makes the connection with the applied science literature in
which noisy limit cycle dynamics are often reduced to a diffusion model for the
phase of the limit cycle.
