this, which allows $0 < \sigma < \frac16$ and is couched in terms suited to our proof of deterministic global existence.

The theorem above implies a probabilistic well-posedness result as well. To state it, we first recall the definition of the Gaussian measure associated to $H^s({\mathbb{T}})$. The measure is given by $${\,\operatorname{d}\!}\mu_s = Z_s^{-1} e^{-\frac12 \| n\|_{H^s}^2} \prod_{x \in {\mathbb{T}}} {\,\operatorname{d}\!}n(x).$$ A typical element in the support of the measure is of the form $$n = n^\omega = \sum_{k \in {\mathbb{Z}}} \frac{g_k(\omega)}{{\langle}k {\rangle}^s} e^{ikx},$$ where $\{g_k(\omega)\}_k$ are independent standard Gaussian random variables. We see that the $n$ is $\omega$-almost-surely in $H^{s-\frac12-}\backslash H^{s-\frac12}$, so the measure is supported on $\bigcap_{r< s - \frac12} H^{r}({\mathbb{T}})$. This leads to the following corollary.

The Zakharov system is almost-surely globally well-posed for initial data in the space $H^s \times H^r$ endowed with the Gaussian probability measure if $$s> \frac{1}{2} \quad \text{and} \quad r >  -\frac{1-s}{2(2-s)}.$$ In particular, we have global existence almost-surely in $H^{\frac12+} \times H^{-\frac1{6}+}$.

The corollary follows by noting that typical data in $H^r$ is of the form $$n_0 = n^\omega_0 = \sum_{k \in {\mathbb{Z}}} \frac{g_k(\omega)}{{\langle}k {\rangle}^{r + \frac12 + \epsilon}} e^{ikx},$$ and thus its Fourier coefficients satisfy $$\sup_k {\langle}k {\rangle}^{ r + 