Abstract : The generalization of classical geometric mechanics $($ including the study of symmetries, Hamiltonian and Lagrangian mechanics, and the Hamilton-Jacobi theory, etc.$)$ to the context of stochastic dynamics has drawn more and more attention in recent decades. One of the important motivations behind some pieces of work related to this field is establishing a framework adapted to the handling of mechanical systems subjected to random perturbations or whose parameters are not precisely determined and are hence modeled as realizations of a random variable. This minisymposium will bring together speakers with diverse but related background, discussing recent developments on general topics of stochastic dynamical systems with Hamiltonian or other geometric structure.
[04102] The Hamilton-Jacobi Theory for Stochastic Hamiltonian Systems on Jacobi Manifold
Format : Talk at Waseda University
Author(s) :
Pingyuan Wei (Beijing International Center for Mathematical Research, Peking University, Beijing 100871, China)
Qiao Huang (Nanyang Technological University)
Abstract : In this talk, we generalize the systems of Hamiltonian diffusions, which were introduced and studied by Bismut, to accommodate arbitrary Jacobi manifolds as phase spaces and general continuous semimartingales as forcing noises. As is well-known, Jacobi structures are the natural generalization of Poisson structure and in particular of symplectic, cosymplectic and Lie-Poisson structures. However, very interesting manifolds like contact and locally conformal symplectic manifolds are also Jacobi but not Poisson. We are interested in the systems such that the phase flows preserve characteristic structures, and develop a Hamilton-Jacobi theory which is regarded as an alternative method for formulating the dynamics. A particular example is the case of thermodynamic dynamics in which we apply our methods on a manifold with its canonical contact form.
[02876] Homogenization of Dissipative Hamiltonian Systems under L\'evy Fluctuations
Format : Talk at Waseda University
Author(s) :
Zibo Wang (Huazhong University of Science and Technology)
Abstract : We study the small mass limit for a class of Hamiltonian systems with multiplicative non-Gaussian L\'evy noise. The limiting equation has a discontinuous noise-induced drift term. First, we show that the momentum in the stochastic Hamiltonian system converges to zero when the kinetic energy has polynomial growth. Then we prove that the stochastic Hamiltonian system with classical kinetic energy converges to the limiting equation in probability with respect to Skorohod topology.
[02877] The stochastic flocking model with far-field degenerate communication
Format : Talk at Waseda University
Author(s) :
Li Lv (Huazhong University of Science and Technology)
Abstract : We reconsider the stochastic kinetic Cucker-Smale model with multiplicative Brownian noise, in which we remove the positive lower bound assumption on the communication. First we prove the emergence of conditional flocking in strong sense. Then, we show that unconditional strong flocking occurs when communication weight decays slowly at the far field. These results imply uniform stability and mean-field limit. In particular, strong stability and mean-field limit in the expectation sense are established in one-dimensional case.
[04106] The Most Likely Transition Path for a Class of Distribution-Dependent Stochastic Systems
Format : Talk at Waseda University
Author(s) :
Wei Wei (Shanghai Jiao Tong University )
Jianyu Hu (Nanyang Technological University)
Abstract : Distribution-dependent stochastic dynamical systems arise widely in engineering and science. We consider a class of such systems which model the limit behaviors of interacting particles moving in a vector field with random fluctuations. We aim to examine the most likely transition path between equilibrium stable states of the vector field. In the small noise regime, we find that the action functional (or rate function) does not involve with the solution of the skeleton equation, which describes unperturbed deterministic flow of the vector field shifted by the interaction at zero distance. As a result, we are led to study the most likely transition path for a stochastic differential equation without distribution-dependency. This enables the computation of the most likely transition path for these distribution-dependent stochastic dynamical systems by the adaptive minimum action method and we illustrate our approach in two examples.
[00862] A parameterization method for quasi-periodic systems with noise
Format : Talk at Waseda University
Author(s) :
Lei Zhang (Dalian University of Technology)
Pingyuan Wei (Beijing International Center for Mathematical Research, Peking University)
Abstract : This work is devoted to studying the existence of invariant tori for a class of quasi-periodically forced systems with stochastic noise, and implementing an efficient method to compute the tori as well as Lyapunov exponents. These systems are skew-product systems driven by small perturbations. Two very common cases of noise included in our treatment are continuous stationary stochastic process and white noise. The existence of invariant tori is established by developing a parameterization method in random setting and applying an elementary fixed point theorem in Banach spaces. Based on this, we describe a numerical algorithm for the computation of them. Moreover, by considering the reducibility for random manifold, we also propose a numerical algorithm for the computation of the corresponding Lyapunov exponents.
[04265] Parametric Resonance for Enhancing the Rate of Metastable Transition
Format : Talk at Waseda University
Author(s) :
Ying Chao (Xi’an Jiaotong University)
Molei Tao (Georgia Inistitute of Technology)
Abstract : In this talk, we will introduce a way to quantify how periodic perturbation can change the rate of metastable transition in stochastic mechanical systems with weak noises. A closed-form explicit expression for approximating the rate change is provided, and the corresponding transition mechanism can also be approximated. Unlike the majority of existing relevant works, these results apply to kinetic Langevin equations with high-dimensional potentials and nonlinear perturbations. They are obtained based on a higher-order Hamiltonian formalism and perturbation analysis for the Freidlin-Wentzell action functional. This tool allowed us to show that parametric excitation at a resonant frequency can significantly enhance the rate of metastable transitions. Numerical experiments for both low-dimensional toy models and a molecular cluster are also provided. For the latter, we show that vibrating a material appropriately can help heal its defect, and our theory provides the appropriate vibration.
[04127] An end-to-end deep learning approach for extracting stochastic dynamical systems with α-stable Lévy noise
Format : Talk at Waseda University
Author(s) :
Cheng Fang (Huazhong University of Science and Technology)
Yubin Lu (Illinois Institute of Technology)
Ting Gao (Huazhong University of Science and Technology)
Jinqiao Duan (Great Bay University)
Abstract : Recently, extracting data-driven governing laws of dynamical systems through deep learning frameworks has gained a lot of attention in various fields. Moreover, a growing amount of research work tends to transfer deterministic dynamical systems to stochastic dynamical systems, especially those driven by non-Gaussian multiplicative noise. However, lots of log-likelihood based algorithms that work well for Gaussian cases cannot be directly extended to non-Gaussian scenarios which could have high error and low convergence issues. In this work, we overcome some of these challenges and identify stochastic dynamical systems driven by $\alpha$-stable Lévy noise from only random pairwise data. Our innovations include: (1) designing a deep learning approach to learn both drift and diffusion coefficients for Lévy induced noise with $\alpha$ across all values, (2) learning complex multiplicative noise without restrictions on small noise intensity, (3) proposing an end-to-end complete framework for stochastic systems identification under a general input data assumption, that is, $\alpha$-stable random variable. Finally, numerical experiments and comparisons with the non-local Kramers-Moyal formulas with moment generating function confirm the effectiveness of our method.
Abstract : In this talk, we will use the framework of stochastic geometric mechanics to describe relations between Schrodinger's variational problem and Onsager's approach to nonequilibrium statistical mechanics. Especially, we will rebuild Onsager's reciprocal relations by introducing Riemannian structures on thermodynamic spaces, and propose a definition of entropy for nonequilibrium systems. This is joint work with Jean-Claude Zambrini.
[00806] Recent progress in spatial isosceles three body problem
Format : Online Talk on Zoom
Author(s) :
Lei Liu (Peking University)
Abstract : Recently, we discovered some new and strong connections between the spatial isosceles three body problem and Symplectic Dynamics. From this perspective, more information can be obtained. Therefore, under the light of Symplectic Dynamics, we obtain plenty of new results. In this talk, we will introduce the isosceles three body problem in symplectic and dynamical point of view, including the dimensional reduction, dynamical analysis, index estimates, open book decomposition and convexity. Finally, I will prove the existence of infinite many oscillate periodic motions in certain parametrical setting.
