AI 中文总结
本文通过路径wise方法证明全次临界区域广义KPZ方程的全局时间适定性，处理$L^\infty$初值初始层，并先以标准KPZ演示技术。
AI 中文摘要
我们通过改编最近应用于广义抛物型Anderson模型[ES26]的策略，提供了全次临界区域广义KPZ方程全局时间适定性的路径wise证明。由于该策略关键依赖于控制上确界范数足以延续解的假设，所需的主要额外成分是对仅具有$L^\infty$初值的gKPZ初始层的处理，而非更常见的$\theta>0$时$C^\theta$初值情形。我们的方法基于围绕确定性轮廓的展开，随后引入积分因子以去除在时间$0$处具有临界尺度的项。出于教学目的，我们首先在标准$(1+1)$维KPZ方程情形下演示证明技术，然后转向计算上更复杂的广义KPZ情形。
英文摘要
We provide a pathwise proof of global-in-time well-posedness for the generalised KPZ equation in the full subcritical regime by an adaptation of the strategy recently applied to the generalised Parabolic Anderson Model in [ES26]. Since this strategy relies crucially on the assumption that control of the supremum norm is sufficient to continue the solution, the main additional ingredient required is a treatment of the initial layer for gKPZ with merely $L^\infty$ initial data, rather than the more usual setting of $C^θ$ initial data with $θ> 0$. Our approach is based on an expansion around a deterministic profile followed by the introduction of an integrating factor in order to remove the terms which have critical scaling at time $0$. For pedagogical purposes, we first demonstrate the proof techniques in the case of the standard $(1+1)$-dimensional KPZ equation before turning to the more computationally involved case of generalised KPZ.
Comments25 pages
