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

This talk provides an overview of Evolutionarily Stable Strategies (ESS) in population games and addresses some common misconceptions found in the classical literature. We focus specifically on Locally Superior Strategies (LSS), which represent a very special case of ESS, examining them in contexts with both finite and infinite action sets. Locally Superior Strategies are particularly significant among evolutionary stability concepts because they act as attractors under the Replicator Dynamic. The main result I will present unifies the finite and infinite frameworks by establishing a necessary and sufficient condition for an ESS to be a LSS.