论文标题
一个空间维度中的随机均质化和HJ方程的有效哈密顿量：双孔案例
Stochastic homogenization and effective Hamiltonians of HJ equations in one space dimension: The double-well case
论文作者
论文摘要
我们考虑使用$ h（p，x，ω）= g（p） +βv（x，ω）$的汉密尔顿方程的汉密尔顿 - 雅各布方程，其中$ v（\ cdot，ω）$是单位幅度的固定和良好的电位。此类方程式的均匀化是在2016年的Armstrong，Tran和Yu的论文中建立的，用于所有连续和强制性的$ G $。在额外的条件下，$ g $是双孔功能（即，它具有两个本地最小值），我们给出了一个新的，完全建设性的均质化证明，为有效的汉密尔顿$ \ edline h $提供了公式。我们使用此公式提供了$ \ Overline H $的图表的完整列表。我们通过分析示例的基本类别来说明我们的结果，突出显示一些推论，这些推论阐明了$ \叠加h $对$ g $，$β$和$ v（\ cdot，ω）$的依赖性，并讨论对偶数对称的三重 - 韦尔·汉密尔顿人的概括。
We consider Hamilton-Jacobi equations in one space dimension with Hamiltonians of the form $H(p,x,ω) = G(p) + βV(x,ω)$, where $V(\cdot,ω)$ is a stationary and ergodic potential of unit amplitude. The homogenization of such equations is established in a 2016 paper of Armstrong, Tran and Yu for all continuous and coercive $G$. Under the extra condition that $G$ is a double-well function (i.e., it has precisely two local minima), we give a new and fully constructive proof of homogenization which yields a formula for the effective Hamiltonian $\overline H$. We use this formula to provide a complete list of the heights at which the graph of $\overline H$ has a flat piece. We illustrate our results by analyzing basic classes of examples, highlight some corollaries that clarify the dependence of $\overline H$ on $G$, $β$ and the law of $V(\cdot,ω)$, and discuss a generalization to even-symmetric triple-well Hamiltonians.
