In this paper we prove that the lack of uniqueness for solutions of the tree dyadic model of turbulence is overcome with the introduction of a suitable noise. The uniqueness is a weak probabilistic uniqueness for all $l^2$-initial conditions and is proven using a technique relying on the properties of the $q$-matrix associated to a continuous time Markov chain.
"Uniqueness for an inviscid stochastic dyadic model on a tree." Electron. Commun. Probab. 18 1 - 12, 2013. https://doi.org/10.1214/ECP.v18-2382