In addition, we assume the regularity estimates $$\label{eq:FormalBd:regularityEst}
\|\{\nabla_{t,x}, \nabla^{2}_{x}\}\varphi\|_{L^{1}_{t,x}}\leq M.$$ The existence of $\varphi$ and $M$ with the above properties is non-trivial. We postpone the discussion of this point, since we only wish to illustrate the main idea of the estimate. We refer the reader to Lemma \[lem:sub-Solution\] for more details on the precise construction required for the purpose of proving Theorem \[thm:MainResult\]. $$\text{Step 3: Gronwall argument}$$ Define an energy $E(t)$ by $$E(t)=\E \int_{\R^{d}}\varphi \rho_{t}^{2}\dx.$$ To compute the time evolution of $E$, use with $S(\rho)=\rho^{2}$ and take the expectation to find $$\label{eq:FormalBd:MainIdentity}
E(t)-E(0)=\E \int_{0}^{t}\int_{\R^{d}}\rho^{2}\left [\partial_{t}\varphi + u \cdot \nabla \varphi-(\Div u)\varphi \right ]+\frac{1}{2}\Psi(\rho)\Delta \varphi \dx\ds,$$ where $\Psi$ is given by the formula $$\Psi(\rho)=2\int_{0}^{\rho}vb^{2}(v)\dv.$$ By Hypothesis \[hyp:noiseFlux\], there exists a deterministic constant $\overline{\rho}$ such that $$\frac{\lambda}{2}\leq \inf_{\rho \geq \overline{\rho}}\frac{\Psi(\rho)}{\rho^{2}} \leq \sup_{\rho \geq \overline{\rho}}\frac{\Psi(\rho)}{\rho^{2}} \leq \Lambda.$$ We now split the integral on the RHS of into the regions where $\rho$ takes values above and below $\overline{\rho}$. The contribution from large densities gives: $$\begin{aligned}
&\E \int_{0}^{t}\int_{\R^{d}}1_{\rho \geq \overline{\rho}}\rho^{2}\left [\partial_{t}\varphi + u \cdot \nabla \varphi-\Div u \varphi+\frac{1}{2}\frac{\Psi(\rho)}{\rho^{2}}\Delta \varphi \right ] \dx\ds \nonumber\\
&\leq \,\E \int_{0}^{t}\int_{\R^{d}}1_{\rho \geq \overline{\rho}}\rho^{2}\left [\partial_{t}\varphi + u \cdot \nabla \varphi+(\Div u)_{-}\varphi+\mathcal{M}^{+}_{\frac{\lambda}{4},\Lambda}(D^{2}\varphi) \right ] \dx\ds \nonumber\\
& \leq \, M \E 