Regularity thresholds for anomalous dissipation and related phenomena in passive scalars

We prove the absence of anomalous dissipation for passive scalars driven by some random autonomous divergence-free vector fields in $\mathbb T^d$.

In dimension $d=2$ we just need continuity almost surely and a mild nondegeneracy condition on the randomness.

In dimension $d\geq 3$ we assume a special geometric structure and almost sure Hölder regularity with a Hölder exponent bigger than $\frac{1}{8}$.

No regularity is assumed on the passive scalar except for boundedness in the initial data.

The proof relies on dimension-theoretic arguments, as opposed to commutator estimates.

A consequence of these results is that the same assumptions prevent (almost surely) many other expected properties of turbulent flows, such as anomalous regularization, the Yaglom-Obukhov-Corrsin law, and Richardson diffusion.