Well posedness and limit theorems for a class of stochastic dyadic models

TL;DR

This paper studies stochastic dyadic models with energy-preserving noise, proving existence, uniqueness, convergence to viscous models under scaling, and dissipation enhancement effects.

Contribution

It introduces new results on weak solutions, convergence rates, a central limit theorem, and dissipation phenomena in stochastic dyadic models.

Findings

Weak solutions are unique in law.

Models converge to viscous models with explicit rates.

Dissipation is enhanced by noise in viscous cases.

Abstract

We consider stochastic inviscid dyadic models with energy-preserving noise. It is shown that the models admit weak solutions which are unique in law. Under a certain scaling limit of the noise, the stochastic models converge weakly to a deterministic viscous dyadic model, for which we provide explicit convergence rates in terms of the parameters of noise. A central limit theorem underlying such scaling limit is also established. In case that the stochastic dyadic model is viscous, we show the phenomenon of dissipation enhancement for suitably chosen noise.