发表机构
University of Rome Tor Vergata; SISSA; École Nationale des Ponts et Chaussées, Institut Polytechnique de Paris, CNRS and INRIA MATHERIALS project-team（罗马托尔维加塔大学; 的里雅斯特国际高等研究学校; 国立路桥学院，巴黎理工学院，法国国家科学研究中心和INRIA MATHERIALS项目组）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过概率空间上的变分方法，利用随机控制表示和Gamma-收敛，识别均匀化拉格朗日量并经勒让德变换得到有效哈密顿量，完整刻画粘性Hamilton-Jacobi方程的均匀化极限，并给出对局部缺陷的稳定性结果。
AI 中文摘要
本文研究了一类具有振荡周期势的粘性 Hamilton--Jacobi 方程的渐近行为，这类方程已在偏微分方程框架下由其他作者研究过，并用有效哈密顿量描述。本文引入了一种新的变分方法，基于经典的涉及布朗运动的随机控制表示，从而将问题重新表述为定义在概率空间上的泛函。核心结果是通过适用于随机过程的 $\Gamma$-收敛过程识别出一个均匀化拉格朗日量，其中滤子和布朗运动被视为变量。有效哈密顿量随后作为该均匀化拉格朗日量的勒让德变换被恢复。分析通过爆破论证和随机校正子的构造提供了下界和上界，从而给出了均匀化极限的完整刻画。该方法允许关于在周期背景中嵌入的一些局部缺陷的稳定性结果。
英文摘要
This work studies the asymptotic behavior of a class of viscous Hamilton--Jacobi equations with oscillatory periodic potentials, already studied by other authors in the PDE framework, and described in terms of an effective Hamiltonian. A novel variational approach is introduced, based on the classical stochastic control representation involving a Brownian motion, which allows the problem to be reformulated in terms of functionals defined on probability spaces. The core result is the identification of a homogenized Lagrangian obtained via a $Γ$-convergence procedure adapted to random processes, where filtrations and Brownian motions are treated as variables. The effective Hamiltonian is then recovered as the Legendre transform of this homogenized Lagrangian. The analysis provides both lower and upper bounds by means of blow-up arguments and the construction of stochastic correctors, yielding a complete characterization of the homogenized limit. This approach allows for a stability result with respect to the addition of some localized defects embedded in the periodic background.
