A stochastic version of an inviscid dyadic model of turbulence, with multiplicative noise, is proved to exhibit energy dissipation in spite of the formal energy conservation. As a consequence, global regular solutions cannot exist. After some reductions, the main tool is the escape bahavior at infinity of a certain birth and death process.
"Anomalous dissipation in a stochastic inviscid dyadic model." Ann. Appl. Probab. 21 (6) 2424 - 2446, December 2011. https://doi.org/10.1214/11-AAP768