  Motivated by the possibility of noise to cure equations of finite-time
blowup, recent work arXiv:2109.09892 by the second and third named authors
showed that with quantifiable high probability, random diffusion restores
global existence for a large class of active scalar equations in arbitrary
dimension with possibly singular velocity fields. This class includes
Hamiltonian flows, such as the SQG equation and its generalizations, and
gradient flows, such as the Patlak-Keller-Segel equation. A question left open
is the asymptotic behavior of the solutions, in particular, whether they
converge to a steady state. We answer this question by showing that the
solutions from arXiv:2109.09892 in the periodic setting converge in Gevrey norm
exponentially fast to the uniform distribution as time $t\rightarrow\infty$.
