Instabilities in Hamiltonian systems

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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.