and $\Lambda$ such that $$\begin{aligned}
\sup_{\rho \in \R} b^{2}(\rho) &\leq \Lambda.\\
\liminf_{|\rho| \to \infty}b^{2}(\rho)&=\lambda.\end{aligned}$$

Note that a typical example of a $B$ satisfying Hypothesis \[hyp:noiseFlux\] is one which either vanishes or admits a power like behavior for small values of the density, and remains bounded from above and below for large values of the density.\
The precise notion of solution to uses the kinetic formulation , which will be discussed in Section \[sec:KES\] below. Namely, we will show existence and uniqueness of so-called kinetic solutions. We temporarily postpone the precise definition and move to a statement of the main result.

\[thm:MainResult\] Let $u$ satisfy Hypotheses \[hyp:drift:renorm\]-\[hyp:drift:subCritical\] and $B$ satisfy Hypothesis \[hyp:noiseFlux\]. For each $\rho_{0} \in L^{1}_{x} \cap L^{\infty}_{x}$, there exists a unique kinetic solution to starting from $\rho_{0}$. Moreover, given two initial datum $\rho_{0,1}$ and $\rho_{0,2}$ along with the associated entropy solutions $\rho_{1}$ and $\rho_{2}$, it holds that for almost all $t \in [0,T]$, $$\label{eq:L1contraction}
\E\int_{\R^{d}}|\rho_{1}(t,x)-\rho_{2}(t,x)|\dx \leq C \int_{\R^{d}}|\rho_{0,1}(x)-\rho_{0,2}(x)|\dx,$$ where the constant $C$ depends on the velocity field $u$, the ellipticity parameters $\lambda, \Lambda$ and the dimension $d$.

We now discuss briefly the strategy of the proof. The general methodology for proving Theorem \[thm:MainResult\] is inspired by a duality method 