My talk is about the effects of small-scale turbulence on large-scale motion by using a stochastic scaling and singular limits. Many works have been done in recent years using the scaling limit in both scalar and vector cases. The second one is characterized by the presence of stretching, which is  an incremental detail over the scalar case. Our (with F. Flandoli) aim in [arXiv:2410.00520] is to understand the stretching mechanism of stochastic models of turbulence acting on a simple model of polymer. Namely, we investigate a scaling limit problem, under suitable intensity assumption. The polymer density equation, initially an SPDE converges (in the first step) weakly to a limit deterministic equation with a new degenerate term with some singular parameter. Recently, in [arXiv:2503.18143] we investigate the singular limit in the spirit of the hydrodynamic limit techniques. One consequence is that the limiting density shows a power-law decay in the polymer length, which is consistent with physical predictions.

The activities mentioned herein were performed in the framework of the project: EU-HORIZON EUROPE ERC-2021-ADG "Noise in Fluids" (NoisyFluid), no. 101053472