Universal long-time properties of Lagrangian statistics in the Batchelor regime and their application to the passive scalar problem

We consider the transport of dynamically passive quantities in the Batchelor regime of a smooth in space velocity field. For the case of arbitrary temporal correlations of the velocity, we formulate the statistics of relevant characteristics of Lagrangian motion. This allows us to generalize many results obtained previously for strain $\delta$ correlated in time, thus answering a question about the universality of these results.