Collaborative Research: Propagation of Dissipation: Stochastic Stabilization in Finite and Infinite Dimensions

Iowa State University, Ames IA

Investigators

Abstract

Randomness is everywhere in modern science and technology. It underlies models of air turbulence, chemical dynamics, and algorithms for big data. Such models are often high dimensional or even infinite dimensional, as in the case of turbulence. Central to understanding these systems is understanding the way both energy and randomness are spread by nonlinear interactions in the dynamics. This research project explores a number of directions that exploit both the geometric and algebraic structures of such systems to better understand these basic transport phenomena. The investigators will work to understand how such interactions can lead to stabilization in systems that are unstable in the absence of randomness. The project includes research participation by graduate, undergraduate, and high-school students. At the high school level, the investigators will also work to keep the students' teachers connected with cutting-edge mathematics, helping them to be more effective and better informed as teachers. In addition, videos chronicling the student research efforts will be made in collaboration with a center for documentary films to further disseminate the experience and broaden the project's impact. This collaborative research project in stochastic analysis and dynamics will explore the propagation of noise and dissipation in stochastic systems and their effects on the existence and structure of stationary states. There is particular emphasis on effects that occur in stochastic partial differential equations, the ability of noise to stabilize unstable systems, and non-equilibrium steady-states in forced systems. Many of the equations to be investigated are physical models (e.g. from fluid mechanics or non-equilibrium statistical physics) while other equations to be studied serve as examples that aid in understanding the underlying mechanisms producing stability or instability in such systems. This work will build on the investigators' previous work in designing Lyapunov functions in the finite-dimensional setting of stochastic ordinary differential equations (SODEs) and establishing practical methods for proving unique ergodicity and convergence to equilibrium in both the finite-dimensional setting of SODEs as well as the infinite-dimensional setting of SPDEs. While the research is anticipated to have immediate implications in fluid mechanics and statistical physics, the techniques that will result from studying such systems will be applicable to a wide family of problems in other areas, such as biology, engineering, physics, and finance, which regularly employ stochastic ordinary and partial differential equations as modeling tools.