  We consider the stochastically driven one dimensional nonlinear oscillator
$\ddot{x}+2\Gamma\dot{x}+\omega^2_0 x+\lambda x^3 = f(t)$ where f(t) is a
Gaussian noise which, for the bulk of the work, is delta correlated (white
noise). We construct the linear response function in frequency space in a
systematic Feynman diagram-based perturbation theory. As in other areas of
physics, this expansion is characterized by the number of loops in the diagram.
This allows us to show that the damping coefficient acquires a correction at
$O(\lambda^2)$ which is the two loop order. More importantly, it leads to the
numerically small but conceptually interesting finding that the response is a
function of the frequency at which a stochastic system is probed. The method is
easily generalizable to different kinds of nonlinearity and replacing the
nonlinear term in the above equation by $\mu x^2$ , we can discuss the issue of
noise driven escape from a potential well. If we add a periodic forcing to the
cubic nonlinearity situation, then we find that the response function can have
a contribution jointly proportional to the strength of the noise and the
amplitude of the periodic drive. To treat the stochastic Kapitza problem in
perturbation theory we find that it is necessary to have a coloured noise.
