AI 中文总结
本文针对马尔可夫扩散理论中显微学到宏观学的过渡，构造协调两种几何描述的核心对象，实现了马尔可夫分析中微观-介观-宏观层级的部分普适化。
AI 中文摘要
在多种不同场景中，人们研究随机过程如何过渡为抛物型偏微分方程，进而过渡为涉及梯度和变分计算的泛函不等式。受梯度力作用的粒子、驱动粒子的场、粒子形成的经验测度、这些测度近似的光滑定律，以及这些定律的可微泛函，构成了扩散的描述层级。该层级中存在一个选择：连续性算子如何将粒子速度转换为测度的演化。本文构造了一个核心对象，用于协调两种此类描述。闭黎曼流形上的光滑正概率密度空间被视为弗雷歇流形，其完整连续余切空间由非常分布构成，正则余切空间由足够正则的非常函数构成。从该空间上源于底流形微分同胚无穷小提升的算子出发，该密度空间的两种不同希尔伯特完备化涵盖了大量相关对象，实现了马尔可夫分析中微观-介观-宏观层级的部分普适化。
英文摘要
In various disparate settings one studies how random processes give way to parabolic partial differential equations and in turn to functional inequalities involving gradients and variational calculus. Particles subject to gradient forces, the fields which move them, the empirical measures which they form, the smooth laws which those measures approximate, and differentiable functionals of these laws constitute a hierarchy of descriptions of diffusion. Intervening on this hierarchy is a choice of how a conintuity operator converts particle velocity to the evolution of measures. Here a central object is constructed mediating two different such descriptions. The space of smooth positive probability densities on a closed Riemannian manifold is treated as a Fréchet manifold whose full continuous cotangent space consists of nonconstant distributions and whose regular cotangent space consists of sufficiently regular nonconstant functions. Beginning from the operator on this space arising as the infinitesimal lift of diffeomorphisms of the base manifold, two different Hilbert completions of this space of densities account for a wide class of objects relevant, leading to a partial universalisation of the microscopic--mesoscopic--macroscopic hierarchy in Markov analysis.
Comments52+3 pages. The main text consists of 49 pages and appendices comprise the remainder
