We study a class of Hamilton–Jacobi partial differential equations in the space of probability measures. In the first part of this paper, we prove comparison principles (implying uniqueness) for this class. In the second part, we establish the existence of a solution and give a representation using a family of partial differential equations with control. A large part of our analysis exploits special structures of the Hamiltonian, which might look mysterious at first sight. However, we show that this Hamiltonian structure arises naturally as limit of Hamiltonians of microscopical models. Indeed, in the third part of this paper, we informally derive the Hamiltonian studied before, in a context of fluctuation theory on the hydrodynamic scale. The analysis is carried out for a specific model of stochastic interacting particles in gas kinetics, namely a version of the Carleman model. We use a two-scale averaging method on Hamiltonians defined in the space of probability measures to derive the limiting Hamiltonian.

我们研究概率测度空间中的一类 Hamilton-Jacobi 偏微分方程。在本文的第一部分，我们证明了这个类的比较原则（暗示唯一性）。在第二部分，我们建立了一个解的存在性，并使用一系列具有控制的偏微分方程给出了一个表示。我们分析的很大一部分利用了哈密顿量的特殊结构，乍一看可能看起来很神秘。然而，我们表明这种哈密顿结构是作为微观模型哈密顿量的极限自然出现的。事实上，在本文的第三部分，我们在流体动力学尺度的涨落理论的背景下非正式地推导出了之前研究的哈密顿量。该分析是针对气体动力学中随机相互作用粒子的特定模型进行的，即卡尔曼模型的一个版本。我们对在概率测度空间中定义的哈密顿量使用两尺度平均方法来推导出极限哈密顿量。