Mixing and transport of passive scalars is an important physical problem. Observed distributions of scalars in convection, the stratosphere and the ocean have heavy tails large probabilities of large fluctuations. Mathematically, heavy tails have been found in random flows with and without chaos. Here we explore the evolution of the Probability Density Functions (PDF) for the random Greens functions for a diffusing passive scalar, which is also advected by a velocity field with stochastic components. These stochastic components are rapidly varying in time and they can be spatially dependent such as shears. Different analytical methods are presented to compute closed-form expressions for these PDFs or their asymptotic limits. Although computed for simple flows, these exact solutions retain mechanisms which are physically relevant. Moreover, these solutions can be effectively used as tests for more general situations where one has to resort to numerical studies of the PDF dynamics because only moment information is available. Interesting new phenomena in the random uniform shear layer will be presented using rigorous asymptotics methods.
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