Multiscale analysis of infinite-dimensional stochastic systems

When studying complex systems, having a simplified description is crucial. Often, this simplification comes from focusing on a select few factors; however, factors that initially seem insignificant can be critical over longer time scales. Thus, a deep understanding of interactions across scales is essential for more effective models of complex systems.

This project will address issues in the asymptotic behavior of infinite-dimensional systems that are governed by stochastic partial differential equations (SPDEs) with multiple scales, focusing on SPDEs with conservation laws—a field that remains largely unexplored for systems of infinite dimensions. The planned work will bridge significant gaps in theory and introduce new approaches to the analysis of these multiscale systems.

Graduate students and postdocs will participate in the research, and the awardee will develop graduate courses and contribute to the broader mathematical community through lectures, organization of events, and editorial service.

Central to this research is the analysis of the Smoluchowski-Kramers diffusion approximation for stochastic systems with an infinite number of degrees of freedom. We aim to prove the validity of the small-mass limit for stochastic damped geometric wave equations, initially concentrating on stochastic wave maps in one dimension and expanding to more complex systems with state-dependent friction.

Both non-local and local friction coefficients will be explored, studying their implications on the trajectories over finite time intervals and on stationary solutions. Further, the project plans to develop an infinite-dimensional version of the classical Friedlin-Wentcell averaging theory for random perturbations of PDEs with conservation laws. This includes constructing SPDEs that live on the level sets of specific functional, establishing the existence invariant measures for these processes, and proving their unique ergodicity and averaging limits.

Through these endeavors, the proposal aims to understand better the long-term effects of small stochastic and deterministic perturbations on complex systems. By achieving a deeper understanding of these interactions, the research not only contributes to the fundamental theories in mathematical physics and applied mathematics but also provides robust tools for addressing similar phenomena in various scientific disciplines.

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.