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This paper considers the damped periodic Korteweg-de Vries (KdV) equation inthe presence of a white-in-time and spatially smooth stochastic source term andstudies the long-time behavior of solutions. We show that the integrals ofmotion for KdV can be exploited to prove regularity and ergodic properties ofinvariant measures for damped stochastic KdV. First, by considering non-trivialmodifications of the integrals of motion, we establish Lyapunov structure byproving that moments of Sobolev norms of solutions at all orders of regularityare bounded globally-in-time; existence of invariant measures follows as animmediate consequence. Next, we prove a weak Foias-Prodi type estimate fordamped stochastic KdV, for which the synchronization occurs in expected value.This estimate plays a crucial role throughout our subsequent analysis. As afirst novel application, we combine the Foias-Prodi estimate with the Lyapunovstructure to establish that invariant measures are supported on $C^\infty$functions provided that the external driving forces belong to $C^\infty$. Wethen establish ergodic properties of invariant measures, treating the regimesof arbitrary damping and large damping separately. For arbitrary damping, wedemonstrate that the framework of `asymptotic coupling' can be implemented fora compact proof of uniqueness of the invariant measure provided thatsufficiently many directions in phase space are stochastically forced. Ourproof is paradigmatic for SPDEs for which a weak Foias-Prodi type propertyholds. Lastly, for large damping, we establish the existence of a spectral gapwith respect to a Wasserstein-like distance, and exponential mixing anduniqueness of the invariant measure follows.

On the long-time statistical behavior of smooth solutions of the weakly damped, stochastically-driven KdV equation