Scale bridging is a central tool in modern-day theoretical science, but our level of understanding of this tool varies. Some scale-bridging situations are completely understood, with clear formal and rigorous arguments; in other situations, like the famous problem of the derivation of Fourier’s law, the arguments are formally clear but without rigorous backup. 

In this talk I will focus on the question what happens to temporal fluctuations when we coarse-grain a system. These fluctuations are described by large-deviation principles; I will show how the variational structure of these large-deviation principles gives rise to a natural coarse-graining method, in which both the high-probability behaviour of the system and the fluctuations are obtained simultaneously and rigorously. The method makes use of the dual formulation of the large-deviation rate functional, and therefore applies to linear and nonlinear systems alike.