The spectrum of the Markov semigroup of a diffusion process, referred to as
the mixing spectrum, provides a detailed characterization of the dynamics of
statistics such as the correlation function and the power spectrum. Stochastic
analysis techniques for the study of the mixing spectrum and a rigorous
reduction method have been presented in the first part (Chekroun et al. 2017)
of this contribution. This framework is now applied to the study of a
stochastic Hopf bifurcation, to characterize 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, i.e., the leaves of the
stable manifold of the limit cycle generalizing the notion of phase, is
essential to understand the effect of the noise and the phenomenon of phase
diffusion. In addition, it is shown that the mixing spectrum has a spectral
gap, even at the bifurcation point, and that correlations decay exponentially
fast. Small-noise expansions of the 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 to understand how the interaction of the noise with the
deterministic dynamics affect the decay of correlations. Numerical results
complement the study of the mixing spectrum at the bifurcation point, revealing
interesting 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 systems. This approach is not limited to
low-dimensional systems and the reduction method presented in part I will be
applied to stochastic models relevant to climate dynamics in Part III.