中科院数学与系统科学研究院
数学研究所
学术报告
动力系统研讨班
报告人：韦屏远 博士(北京国际数学研究中心,北京大学)
题 目：A Brief Introduction to Stochastic Hamiltonian Systems on Jacobi Manifold
时 间：2023.07.12（星期三）15:00-16:00
地 点：数学院南楼N913
摘 要：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. 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. We show that such systems preserve characteristic structures and develop a stochastic version of Hamilton–Jacobi theory.
(We remake that both Jacobi and Poisson manifolds were introduced by Lichnerowicz. Jacobi ones are regarded as the natural generalization of Poisson ones and in particular of symplectic, cosymplectic and Lie–Poisson structures. However, very interesting manifolds like contact manifolds and locally conformal symplectic manifolds are also Jacobi and they are not Poisson.)
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报告人：张琦 博士(北京雁栖湖应用数学研究院/清华大学丘成桐数学中心)
题 目：Macroscopic coarse-grained dynamics: from most probable path to macroscopic fluctuation
时 间：2023.07.12（星期三）16:00-17:00
地 点：数学院南楼N913
摘 要：Coarse-graining is the process of describing a high-dimensional and complex microscopic dynamics by an macroscopic dynamics. In this talk, we consider the overdamped Langevin dynamics as a microscopic dynamics model. The density and current, as macroscopic observables, are derived from the ensemble-averaged empirical density and current using conditional expectations and the law of large numbers. By employing the the Onsager-Machlup action functional, we determine the most probable path as the macroscopic dynamics. Additionally, we establish a large deviations principle for the ensemble-averaged empirical density and current. Finally, we compare these results with the hydrodynamical limit of stochastic lattice gases, as first considered by Guo, Papanicolaou, and Varadhan.
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