Instabilities in Hamiltonian systems

哈密顿系统的不稳定性一直是动力系统领域的核心问题。这个问题是由天体力学（太阳系的稳定性）和统计物理学（热力学定律的微观基础）引起的。其中一个重要的问题是Arnold扩散，即完全可积系统的典型扰动是否表现出拓扑不稳定性。这个问题既困难又深刻，因为几乎可积系统（如KAM和Nekhoroshev理论）具有显著的稳定性。在Vadim Kaloshin和其他合作者的帮助下，我们在两个半自由度的光滑哈密顿系统中回答了这个问题。然而，我们对不稳定性的理解仍然非常有限。我提出的研究从两个不同的方向加深了我们的知识。1. 拓扑不稳定性在更高的自由度，并在分析范畴。与Vadim Kaloshin一起，我们提出了一个计划，利用对低维结构的约简来研究高自由度的Arnold扩散。我们希望能在流畅类中完整地回答Arnold的问题。另一方面，阿诺德对分析系统提出了他最初的问题，在这个领域，人们所知甚少。有了我们在光滑情况下的知识，我建议从更简单的模型开始，转向分析情况。2. 不稳定轨道的随机描述。“阿诺德扩散”这个术语之所以被创造出来，是因为基于数值证据，不稳定体表现出一种随机行走的行为，就像扩散轨道一样。我建议证明证明扩散方面的随机极限定理。从哲学上讲，这与“确定性随机性”的概念是一致的，即随机行为可以从完全确定性的系统中出现。研究将在两个方向上进行：在不稳定模型中，如先验不稳定模型，可以嵌入通常双曲层，其动力学与随机动力系统共轭；在慢快系统中，扩散过程的极限可以被证明。在提出本研究的同时，我们也提高了对基础理论的认识，特别是弱鉴理论。这些结果将应用于相关领域，如正则和随机汉密尔顿-雅可比方程。

Instabilities in Hamiltonian systems has been a central question in dynamical systems, since the founding of the field. This question has both been motivated by celestial mechanics (stability of the solar system) and by statistical physics (micro foundation of the thermodynamic laws). Among them, one important question is the Arnold diffusion, which asks whether a typical perturbation of a completely integrable system exhibit topological instability. This question is both difficult and deep, due to the remarkable stability enjoyed by nearly integrable systems, such as KAM and Nekhoroshev theory. With Vadim Kaloshin and other collaborators, we have been key contributors towards answering this question in two and a half degree of freedom, for smooth Hamiltonian systems. However, our understanding of the instability is still very limited. My proposed research deepens our knowledge in two separate directions. 1. Topological instability in higher degrees of freedom, and in the analytic category. With Vadim Kaloshin, we proposed a plan to study Arnold diffusion in higher degrees of freedom, using reduction to lower dimensional structures. We hope to fully answer Arnold's question in the smooth category. On the other hand, Arnold asked his original questions for analytic systems, where very little is known. With our knowledge in the smooth case, I propose to tack the analytic case, starting from simpler models. 2. Stochastic description of unstable orbit. The term "Arnold diffusion" was coined because, based on numerical evidence, the unstable exhibit a random-walk-like behavior, just like a diffusion orbit. I propose to prove stochastic limit theorems which justify the diffusion aspect. Philosophically, this aligns with the idea of "deterministic randomness", where random behavior can emerge from fully deterministic systems. The research will be pursued in two directions: in models of instability such as the a priori unstable model, one can embed a normally hyperbolic lamination, on which the dynamics is conjugate to a random dynamical systems; in slow fast system, where limit to a diffusion process can be proven. With the propose the research, we also advance our knowledge in the underlying theory, in particular weak KAM theory. These results will be applied to related fields, such as regular and random Hamilton-Jacobi equations.