We study the derivative nonlinear wave equation \( - \partial_{tt} u + \Delta u = |\nabla u|^2 \) on \( \mathbb{R}^{1+3} \). The deterministic theory is determined by the Lorentz-critical regularity \( s_L = 2 \), and both local well-posedness above \( s_L \) as well as ill-posedness below \( s_L \) are known. In this paper, we show the local existence of solutions for randomized initial data at the super-critical regularities \( s\geq 1.984\). In comparison to the previous literature in random dispersive equations, the main difficulty is the absence of a (probabilistic) nonlinear smoothing effect. To overcome this, we introduce an adaptive and iterative decomposition of approximate solutions into rough and smooth components. In addition, our argument relies on refined Strichartz estimates, a paraproduct decomposition, and the truncation method of de Bouard and Debussche.

中文翻译：

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微分非线性波动方程的几乎肯定的局部适定性

我们研究了 \( \mathbb{R}^{1+3} \) 上的导数非线性波动方程 \( - \partial_{tt} u + \Delta u = |\nabla u|^2 \)。确定性理论由洛伦兹临界正则性\( s_L = 2 \) 确定，并且\( s_L \) 以上的局部适定性以及\( s_L \) 以下的不适定性都是已知的。在本文中，我们展示了在超临界规律 \( s\geq 1.984\) 处随机初始数据解的局部存在性。与之前关于随机色散方程的文献相比，主要的困难是缺乏（概率）非线性平滑效应。为了克服这个问题，我们引入了近似解的自适应和迭代分解为粗糙和平滑的组件。此外，我们的论点依赖于精炼的 Strichartz 估计、副产物分解、