Strictness characterization for divergence-free advection-diffusion flows

Let $u$ be a divergence-free vector field, let $mu aftrightarrow\nu$, and let $\nu$ be symmetric decreasing. For $t\geq 0$, write $\mathcal{P}_s^u\mu$ for the measure evolved by the advection-diffusion flow generated by $u$ up to time $s$. Then the following are equivalent:

Strictness characterization.

$$\text{For every }s\in[0,t],\qquad e^{s\Delta}\nu\preceq\mathcal{P}_s^u\mu.$$

The measure $\mu$ is a translate of $\nu$, and there exist functions $v:[0,t]\to\mathds{R}^d$ and $A:[0,t]\to\mathds{R}^{d\times d}$ such that

$$u(s,x)=v(s)+A(s)x,$$

with

$$A(s)^\top=-A(s)$$

for every $s\in[0,t]$ and $x\in\mathds{R}^d$. Equivalently, equality can persist up to time $t$ only when the advecting flow consists of translations and time-dependent solid-body rotations. The claim gives the anticipated equality case for the concentration comparison; its validity is not established in the supplied text.