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

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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.