In this VICI proposal we bring three concepts together: gradient flows, large-deviation principles, and the chal- lenge of bridging scales. Gradient flows are a class of differential equations that model a wide range of evolving systems. The gradient-flow structure is used extensively to address in a unified way questions of well-posedness, stability of solutions, existence of travelling waves and other coherent structures, parameter dependence, large-time behaviour, and many others.
The first aim of this proposal is to use the gradient-flow structure to address a key challenge in many scientific and technological problems: the bridging of spatial and temporal scales. Building on variational and geometric properties of gradient flows we will develop new methods to scale up nonlinear systems and characterize behaviour of these systems that emerges at larger scales.
As the second aim we will explore the connnections between gradient-flow structures and stochastic processes. This is suggested by recent work, in which we constructed a new connection between a specific gradient flow and the large-deviation rate functional of a stochastic particle system. This connection explains the origin of the gradient-flow structure, complements its analysis, and introduces new principles for the modelling of real-world systems. This work suggests that other gradient flows may similarly arise from stochastic processes, and we will explore the full extent and implications of this suggestion.
The outcome of this proposal will be new theory and methods for the bridging of scales, new tools for the modelling and analysis of gradient flows, and new understanding of the intertwining of gradient flows, large-deviation principles, and scale bridging. We shall work in a mathematically rigorous way, using methodology from the fields of nonlinear differential equations, variational calculus, homogenization, and probability theory. Key challenges are the characterization of fine-scale structure and the interaction between fluctuations, nonlinearity, and the gradient-flow structure.