  This paper considers the damped periodic Korteweg-de Vries (KdV) equation in
the presence of a white-in-time and spatially smooth stochastic source term and
studies the long-time behavior of solutions. We show that the integrals of
motion for KdV can be exploited to prove regularity and ergodic properties of
invariant measures for damped stochastic KdV. First, by considering non-trivial
modifications of the integrals of motion, we establish Lyapunov structure by
proving that moments of Sobolev norms of solutions at all orders of regularity
are bounded globally-in-time; existence of invariant measures follows as an
immediate consequence. Next, we prove a weak Foias-Prodi type estimate for
damped stochastic KdV, for which the synchronization occurs in expected value.
This estimate plays a crucial role throughout our subsequent analysis. As a
first novel application, we combine the Foias-Prodi estimate with the Lyapunov
structure to establish that invariant measures are supported on $C^\infty$
functions provided that the external driving forces belong to $C^\infty$. We
then establish ergodic properties of invariant measures, treating the regimes
of arbitrary damping and large damping separately. For arbitrary damping, we
demonstrate that the framework of `asymptotic coupling' can be implemented for
a compact proof of uniqueness of the invariant measure provided that
sufficiently many directions in phase space are stochastically forced. Our
proof is paradigmatic for SPDEs for which a weak Foias-Prodi type property
holds. Lastly, for large damping, we establish the existence of a spectral gap
with respect to a Wasserstein-like distance, and exponential mixing and
uniqueness of the invariant measure follows.
