  We present the global-in-time existence of strong solutions and its
large-time behavior for the Kuramoto-Sakaguchi equation with inertia. The
equation describes the evolution of the probability density function for a
large ensemble of Kuramoto oscillators under the effects of inertia and
stochastic noises. We consider a perturbative framework around the equilibrium,
which is a Maxwellian type, and use the classical energy method together with
our careful analysis on the macro-micro decomposition. We establish the
global-in-time existence and uniqueness of strong solutions when the initial
data are sufficiently regular, not necessarily close to the equilibrium, and
the noise strength is also large enough. For the large-time behavior, we show
the exponential decay of solutions towards the equilibrium under the same
assumptions as those for the global regularity of solutions.
