  We study the mixing and dissipation properties of the advection-diffusion
equation with diffusivity $0 < \kappa \ll 1$ and advection by a class of random
velocity fields on $\mathbb T^d$, $d=\{2,3\}$, including solutions of the 2D
Navier-Stokes equations forced by sufficiently regular-in-space, non-degenerate
white-in-time noise. We prove that the solution almost surely mixes
exponentially fast uniformly in the diffusivity $\kappa$. Namely, that there is
a deterministic, exponential rate (independent of $\kappa$) such that all
mean-zero $H^1$ initial data decays exponentially fast in $H^{-1}$ at this rate
with probability one. This implies almost-sure enhanced dissipation in $L^2$.
Specifically that there is a deterministic, uniform-in-$\kappa$, exponential
decay in $L^2$ after time $t \gtrsim |\log \kappa|$. Both the $O(|\log
\kappa|)$ time-scale and the uniform-in-$\kappa$ exponential mixing are optimal
for Lipschitz velocity fields and, to our knowledge, are the first rigorous
examples of velocity fields satisfying these properties (deterministic or
stochastic). This work is also a major step in our program on scalar mixing and
Lagrangian chaos necessary for a rigorous proof of the Batchelor power spectrum
of passive scalar turbulence.
