Degenerate diffusions in finite and infinite dimensions: smoothing and convergence

Iowa State University, Ames IA

Investigators

Abstract

Dynamics that evolve randomly in time are used to describe the behavior of financial markets, turbulence, and complex biological systems. Furthermore, they play a key role in big data algorithms. In many of these contexts, randomness enters the system only through a limited number of directions in space. This sparsity of randomness and the often high or infinite-dimensional nature of the equations lead to numerous mathematical challenges, from obtaining basic structural properties of to establishing finer aspects of the dynamics. This project will explore how the limited randomness interacts with nonlinearity to produce smoothing, leading to criteria for the existence of solutions and equilibria. The work will focus on fluid models with random forcing terms, where very little is known about such behavior. Finer properties of the stochastic dynamics will also be studied, especially how fast the system converges to equilibrium as certain parameters in the system are removed. The aim in this context is to understand the anomalous dissipation phenomenon in fluids. The planned work will involve postdoctoral researchers, graduate students, and undergraduate students. Further synergistic activities will include an online cross-university research reading group and a probability reading group at Iowa State University. This project encompasses several topics at the interface of stochastic analysis and dynamical systems. Diffusions with degenerate noise in both high and infinite dimensions will be studied. A key goal is to understand how noise interacts with nonlinearities to produce smoothing, in the sense that the dynamics belongs to an improved Sobolev space for positive times, and convergence, in the sense that the system settles into a unique statistically steady state for large times. Such understanding is fundamental for deducing basic structural properties of solutions (e.g. large-time existence/ergodicity in the relevant topology), yet it is absent in many important models in turbulence and statistical mechanics. Finer properties of such systems (e.g. classification of irregular points, the meaning of "hypoellipticty" in infinite dimensions, and the anomalous dissipation phenomenon) will also be studied. Many of the equations to be investigated are physical models (e.g. the two-dimensional stochastically forced and damped Euler equations or second-order Langevin dynamics), while others are simplified models to help build understanding in the motivating equations. This work will build on previous research of the awardee by developing methods to analyze the systems for both large and bounded values of the phase space. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.