The spectrum of the generator (Kolmogorov operator) of a diffusion process,
referred to as the Ruelle-Pollicott (RP) spectrum, provides a detailed
characterization of correlation functions and power spectra of stochastic
systems via decomposition formulas in terms of RP resonances. Stochastic
analysis techniques relying on the theory of Markov semigroups for the study of
the RP spectrum and a rigorous reduction method is presented in Part I. This
framework is here applied to study a stochastic Hopf bifurcation in view of
characterizing the statistical properties of nonlinear oscillators perturbed by
noise, depending on their stability. In light of the H\"ormander theorem, it is
first shown that the geometry of the unperturbed limit cycle, in particular its
isochrons, is essential to understand the effect of noise and the phenomenon of
phase diffusion. In addition, it is shown that the spectrum has a spectral gap,
even at the bifurcation point, and that correlations decay exponentially fast.
Explicit small-noise expansions of the RP eigenvalues and eigenfunctions are
then obtained, away from the bifurcation point, based on the knowledge of the
linearized deterministic dynamics and the characteristics of the noise. These
formulas allow one to understand how the interaction of the noise with the
deterministic dynamics affect the decay of correlations. Numerical results
complement the study of the RP spectrum at the bifurcation, revealing useful
scaling laws. The analysis of the Markov semigroup for stochastic bifurcations
is thus promising in providing a complementary approach to the more geometric
random dynamical system approach. This approach is not limited to
low-dimensional systems and the reduction method presented in part I is applied
to a stochastic model relevant to climate dynamics in part III
