  In this work we study permanence of hyperbolicity for autonomous differential
equations under nonautonomous random/stochastic perturbations. For the linear
case, we study robustness and existence of exponential dichotomies for
nonautonomous random dynamical systems. Next, we establish a result on the
persistence of hyperbolic equilibria for nonlinear differential equations. We
show that for each nonautonomous random perturbation of an autonomous
semilinear problem with a hyperbolic equilibrium there exists a bounded
\textit{random hyperbolic solution} for the associated nonlinear nonautonomous
random dynamical systems. Moreover, we show that these random hyperbolic
solutions converge to the autonomous equilibrium. As an application, we
consider a semilinear differential equation with a small nonautonomous
multiplicative white noise, and as an example, we apply the abstract results to
a strongly damped wave equation.
