Conjecture on Gaussian absolute continuity and Gibbs measures in Benjamin–Ono Birkhoff coordinates

Let $s>-1/2$, let $h^{s+\frac{1}{2}}$ denote the sequence space in which the Birkhoff coordinates take values, and let $\rho_s$ be the invariant measure on $H^s_0(\mathbb{T})$ obtained as the pullback of a measure $4\mu$ on $h^{s+\frac{1}{2}}$ by the Birkhoff map. Let $\theta$ be a standard complex Gaussian, let $4\zeta_n^*=n^{-1}$, and for $s<1/2$ define formally the densities

$$G_N(\zeta)=\chi\left(\sum_{k=1}^{N}n|\zeta_n|^2-c_N\right)\exp\left(\sum_{k=1}^N\left(\sum_{k_1=k}^N|\zeta_{k_1}|^2\right)^2\right),$$

where $4\chi:\mathbb{R}\to\mathbb{R}$ is continuous with compact support and $c_N\approx\log(N)$ is a renormalisation constant.

Gaussian absolute-continuity conjecture. If $4\mu$ is a suitable Gaussian measure on $h^{s+\frac{1}{2}}$, then the corresponding measure $4\rho_s$ is absolutely continuous with respect to a suitable Gaussian measure on $H^s_0(\mathbb{T})$. More specifically, the image under the Birkhoff map of the Gibbs measure defined in the cited work should be the measure on $h^s$ obtained as the limit, as $N\to\infty$, of $G_N(\zeta)\mu(d\zeta)$. A similar procedure may lead to the measures considered in the cited works.

This conjecture proposes a probabilistic description of invariant measures for the Benjamin–Ono equation in Birkhoff coordinates. The source does not state a resolution, so the conjecture remains open.