The probabilistic scaling conjecture for nonlinear Schrödinger equations

Let $r$ and $d$ be positive integers, and let $s_p=-\frac{1}{2r}$ be the critical threshold for probabilistic scaling. Consider the nonlinear Schrödinger equation referred to as $(\mathrm{nlsex})$ and random initial data

$$u_{\mathrm{in}}=\sum_{k\in\mathbb{Z}^d}\frac{g_k}{\langle k\rangle^{\alpha}}e^{ik\cdot x},\qquad \alpha=s+\frac{d}{2},$$

where the $g_k$ are independent Gaussian random variables satisfying $\mathbb{E}g_k=0$ and $\mathbb{E}|g_k|^2=1$. Almost surely, $u_{\mathrm{in}}$ belongs to $H^{s-}(\mathbb{T}^d)$. Probabilistic scaling conjecture. If $s>s_p$, then $(\mathrm{nlsex})$ is almost surely locally well-posed for these random initial data, in the sense of the theorem described in the source. This conjecture proposes almost-sure local well-posedness at every regularity above the probabilistic scaling threshold, which is independent of the dimension. The heuristic motivating it uses a square-root gain from Gaussian cancellations, while controlling high–low interactions is identified as a principal difficulty; the source does not report a resolution.