Mathematical Sciences: Large Noise Asymptotics & Numerics of Degenerate Stochastic Differential Systems

本文研究了退化大噪声扰动下非线性动力系统的渐近性，特别是不变测度、Lyapunov数和旋转数的渐近性。要研究的噪声的种类是那些使系统的行为从不稳定到稳定急剧变化的噪声。李雅普诺夫数和旋转数的数值格式将通过研究由这些格式生成的退化马尔可夫链的遍历性质来发展。计算机模拟和物理实验应伴随研究。Volker Wihstutz教授将研究随机力（可能具有高能量，如地震，波浪和风）影响下的动力系统，其中影响不是均匀分布在所有维度（简并）。系统在极高能量理论情况下的统计行为（达到的状态和频率）揭示了系统在正常条件下的行为。特别令人感兴趣的问题是，哪种类型的随机力（噪音）可以稳定一个不稳定的系统，如倒立摆或高层建筑。理论分析应辅以计算机模拟和物理实验。为此，需要制定同时考虑到情况的随机性和简并性的数值方案。

This research will investigate the asymptotics of nonlinear dynamical systems perturbed by degenerate large noise, especially the asymptotics of invariant measures, Lyapunov and rotation numbers. The classes of noise to be studied are those which change the system behavior drastically from unstable to stable. Numerical schemes for the Lyapunov and rotation numbers shall be developed by investigating the ergodic properties of the degenerate Markov chains generated by the schemes. Computer simulations and physical experiments shall accompany the research. Professor Volker Wihstutz will investigate dynamical systems under the influence of random forces (with possibly high energy, such as earthquake, waves and wind), where the impact is not evenly distributed over all dimensions (degeneracy). The statistical behavior of the system in the theoretical situation of extremely high energy (which states are attained and how often) sheds light on the system's behavior under normal conditions. Of particular interest is the question of which types of random forces (noise) can stabilize an otherwise unstable system such as an inverted pendulum or a high rise building. The theoretical analysis shall be accompanied by computer simulations and physical experiments. For this purpose numerical schemes need to be developed which take into account both the randomness and the degeneracy of the situation.