was inspired by Lebowitz-Rose-Speer [@LRS]. See also [@Bour3] for a similar argument for the nonlinear Schrödinger equation.

Thus, global existence is known deterministically in $H^{\frac9{13}+}({\mathbb{T}}) \times L^2({\mathbb{T}}) $ and probabilistically in $H^{\frac12-}({\mathbb{T}})  \times H^{-\frac12-}({\mathbb{T}}) $. The object of this note is to prove a global existence result which, at least partially, fills this gap. Note that one cannot simply use a preservation of regularity argument to conclude some sort of global existence between these spaces. This is because the support of the Gibbs measure is in $H^{\frac12-}\backslash H^{\frac12} \times H^{-\frac12-} \backslash H^{-\frac12}$; that is, the probabilistic result does not give any clear insight into the dynamics in smoother spaces.

The idea of our proof is to combine the high-low decomposition method of Bourgain [@Bour4] with the I-method of Colliander-Keel-Staffilani-Takaoka-Tao [@CKSTT]. This combination has appeared in the work of Bourgain for the quintic Schrödinger equation on ${\mathbb{T}}$ [@Bour5]. We note that we were motivated in part by the probabilistic global existence result of Colliander-Oh [@CoOh], which employed the high-low method. We now give an outline of the method.

The high-low method works by breaking the initial data into high-frequency and low-frequency parts, at some cut-off value $N$. One then solves the 