the full story, since for a fixed realization of the noise, the solution to cannot be more regular than the initial $\rho_{0}$. In the non-linear setting , this intuition is less clear, since the equation for $\E[\beta(\rho)]$ is not closed. Instead, it is coupled to an equation for a different observable $\E[\Psi(\rho)]$, making it less transparent whether $\E[\beta(\rho)]$ is any better behaved than $\rho$ itself. Nonetheless, we show that, in some sense, the role of linear parabolic equations in the analysis of may be replaced by fully non-linear parabolic equations in order to capture a regularizing effect of the noise in the non-linear setting of . This is implemented via a duality appraoch that we discuss below. We now state our results more precisely.\
The hypotheses on the deterministic drift $u$ and the noise flux $B$ are as follows.

\[hyp:drift:renorm\]\[Renormalization\] The drift $u$ belongs to $L^{1}_{t}(W^{1,1}_{x})$ and $\Div u \in L^{r}_{t,x}$ for some $r>1$.

\[hyp:drift:subCritical\]\[Sub-criticality\] The drift $u$ belongs to $L^{\infty}_{t,x}$ and $(\Div u)_{-} \in L^{q}_{t,x}$ for some $q>d+2$. Moreover, there exists an $R>0$ such that $u$ is supported in $[0,T] \times B_{R}$.

\[hyp:noiseFlux\]\[Asymptotic ellipticity\] The function $B$ belongs to $C^{3}(\R;\R)$ with derivative $b(\rho):=B'(\rho)$. Moreover, there exist strictly positive $\lambda$ 