Fluctuation-response relation in turbulent systems

We address the problem of measuring time-properties of Response Functions (Green functions) in Gaussian models (Orszag-McLaughlin) and strongly non-Gaussian models (shell models for turbulence). We introduce the concept of halving time statistics to have a statistically stable tool to quantify the time decay of Response Functions and Generalized Response Functions of high order. We show numerically that in shell models for three dimensional turbulence Response Functions are inertial range quantities. This is a strong indication that the invariant measure describing the shell-velocity fluctuations is characterized by short range interactions between neighbouring shells.