along one spatial dimension. The constraint on norm of the Schroödinger part in $L^2$ is necessary on ${\mathbb{T}}^2$ to ensure that the Hamiltonian constrols the relevant Sobolev norm.

In the one-dimensional case, Bourgain gave the following probabilistic global well-posedness result for :

\[BourThm\] There are Sobolev exponents $0 < \sigma < s < \frac12 < \mu <1$ such that the Zakharov system is well-posed for data $(u_0, n_0, n_1)$ satisfying $$u_0 \in H^s({\mathbb{T}}), \qquad \sup_k |k|^{\mu} | {\widehat}{u_0}(k)| < \infty,$$ $$\sup_k |k|^{-\sigma} | {\widehat}{n_0}(k)| < \infty, \qquad \sup_k | k |^{-\sigma - 1} | {\widehat}{n_1}(k)| < \infty.$$ Moreover, the system is almost surely globally well-posed with respect to the normalized Gibbs measure associated with the Zakharov equation, which is supported on $ \bigcap_{s < \frac12}\Bigl(H^s \times H^{s-1} \times H^{s-2}\Bigr)$.

An examination of Bourgain’s proof reveals that one should take $\sigma$ very close to zero, and $s$ and $\mu$ very close to one half. The almost-sure global existence was established by recasting the equation as an infinite dimensional Hamiltonian system for the spatial Fourier coefficients. The global well-posedness holds almost surely with respect to the Gibbs measure, a Sobolev space probability measure based on the Hamiltonian of the equation. This approach 