The current paper is concerned with the global
dynamics of a class of nonlinear oscillators driven by real or
white noises, of which a typical example is a shunted
Josephson junction exposed to some random medium.
Applying random dynamical systems theory,
it is shown that a driven oscillator in the class under consideration
with a tempered real noise
has a one-dimensional global random attractor
provided that the damping is not too small. Moreover, restricted
to the global attractor, the oscillator induces a random dynamical
system on $S^1$. It is then shown that there is a rotation number
associated to the oscillator which characterizes the speed at
which
the solutions of the oscillator move around the global attractor. The results
extend the existing ones for time periodic and quasi-periodic
Josephson junctions and can be applied to Josephson junctions
driven by white noises.