Stretching of polymers and turbulence: Fokker Planck equation, special stochastic scaling limit and stationary law

The aim of this work is understanding the stretching mechanism of stochastic models of turbulence acting on a simple model of dilute polymers. We consider a turbulent model that is white noise in time and activates frequencies in a shell N ≤ |k| ≤ 2N and investigate the scaling limit as N → ∞, under suitable intensity assumption, such that the stretching term has a finite limit covariance. The polymer density equation, initially an SPDE, converges weakly to a limit deterministic equation with a new term. Stationary solutions can be computed and show power law decay in the polymer length.