Uniqueness for an inviscid stochastic dyadic model on a tree

TL;DR

This paper demonstrates that adding appropriate noise to an inviscid stochastic dyadic model on a tree ensures solution uniqueness, using Markov chain techniques to establish weak probabilistic uniqueness for all initial conditions.

Contribution

It introduces a noise-based method to achieve uniqueness in a previously non-unique inviscid stochastic dyadic turbulence model on a tree.

Findings

Noise ensures solution uniqueness in the model

Weak probabilistic uniqueness holds for all initial conditions

Markov chain techniques are effective in the proof

Abstract

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.