论文标题
部分可观测时空混沌系统的无模型预测
Hamilton-Jacobi equations from mean-field spin glasses
论文作者
论文摘要
我们给汉密尔顿 - 雅各比方程赋予粘度意义上的平均场自旋玻璃模型，并建立相应的良好性。最初定义在单调概率度量集中的集合，可以通过等距来解释这些方程式，以在无限二维闭合凸锥上定义，并在希尔伯特空间中具有空的内部。我们证明了比较原理，以及有限维近似的收敛性，可提供溶液的存在。在其他凸度条件下，我们表明该解决方案可以用hopf-lax公式的版本或锥体上的hopf公式表示。以前，考虑了两个溶液概念，一种直接定义为Hopf-最大公式，另一种是有限维近似的限制。事实证明，它们可以描述各种平均场自旋玻璃模型中的极限自由能。这项工作表明，这两种解决方案是粘度解决方案。
We give a meaning to the Hamilton--Jacobi equation arising from mean-field spin glass models in the viscosity sense, and establish the corresponding well-posedness. Originally defined on the set of monotone probability measures, these equations can be interpreted, via an isometry, to be defined on an infinite-dimensional closed convex cone with an empty interior in a Hilbert space. We prove the comparison principle, and the convergence of finite-dimensional approximations furnishing the existence of solutions. Under additional convexity conditions, we show that the solution can be represented by a version of the Hopf--Lax formula, or the Hopf formula on cones. Previously, two notions of solutions were considered, one defined directly as the Hopf--Lax formula, and another as limits of finite-dimensional approximations. They have been proven to describe the limit free energy in a wide class of mean-field spin glass models. This work shows that these two kinds of solutions are viscosity solutions.
