The nonsmooth Kraichnan model \cite{Kra1968} is a linear stochastic transport model in which the velocity field is Gaussian, white in time, incompressible, isotropic, and spatially rough. Introduced as a toy model for turbulent transport, it allows explicit computations and has become a benchmark for predictions such as Richardson pair dispersion and intermittency.

In mathematics, it stands out as one of the few transport models exhibiting both well-posedness and spontaneous stochasticity, as shown in the early 2000s by Le Jan and Raimond \cite{LeJRai2002}, and independently by E and Vanden-Eijnden \cite{EVan2001}. More recently, renewed interest has followed rigorous results on anomalous dissipation and anomalous regularization (e.g. \cite{Row2024,GalGroMau2024,DriGalPap2025}), as well as related regularization-by-noise results (e.g. \cite{CogMau2023,BagGalMau2025}).

In this talk, we will review these mathematical developments and highlight key open problems in stochastic turbulent transport.