solution to $\eqref{eq:ContEq}$ concentrates mass at the origin in finite time.

In an influential paper of Flandoli/Gubinelli/Priola [@MR2593276], it is proved that the conditions required for well-posedness of transport equations may be improved under a white in time perturbation of the underlying velocity field of the form $u \to u+\sigma \dot{W}$. In the original article, the authors considered the equation $$\label{eq:stochTrans}
\partial_{t}\rho+\nabla\rho \circ (u+\sigma\dot{W})=0 \quad \text{on} \quad [0,T] \times \R^{d},$$ where the product is understood in the Stratonovich sense. The transport equation corresponding to $\sigma=0$ is, in a sense, dual to the continuity equation . The issue of mass concentration for the continuity equation is in turn dual to the question of uniqueness for transport equations, and the counterexample above persists. A key result of [@MR2593276] is that if $u \in C_{t}(C^{\alpha}_{x})$, then the condition on $\Div u$ required for uniqueness may be significantly weakened to $\Div u \in L^{2}_{t,x}$. Moreover, if the initial datum posess BV regularity, then the condition on $\Div u$ may be dropped entirely. The main workhorse for these results is a study of the stochastic flow corresponding to the SDE $$\label{eq:stochChar}
\dot{X}_{t}=u(t,X_{t})+\sigma\dot{W}_{t},$$ where the Hölder continuity of $u$ allows to build a differentiable stochastic flow 