  We study the decay rate of the Loschmidt echo or fidelity in a chaotic system
under a time-dependent perturbation $V(q,t)$ with typical strength
$\hbar/\tau_{V}$. The perturbation represents the action of an uncontrolled
environment interacting with the system, and is characterized by a correlation
length $\xi_0$ and a correlation time $\tau_0$. For small perturbation
strengths or rapid fluctuating perturbations, the Loschmidt echo decays
exponentially with a rate predicted by the Fermi Golden Rule, $1/\tilde{\tau}=
\tau_{c}/\tau_{V}^2$, where typically $\tau_{c} \sim \min[\tau_{0},\xi_0/v]$
with $v$ the particle velocity. Whenever the rate $1/\tilde{\tau}$ is larger
than the Lyapunov exponent of the system, a perturbation independent Lyapunov
decay regime arises. We also find that by speeding up the fluctuations (while
keeping the perturbation strength fixed) the fidelity decay becomes slower, and
hence, one can protect the system against decoherence.
