$$\label{eq:lowVelBdsonKM}
\int_{0}^{T}\iint_{\R^{d}\times [-R,R]}m_{\epsilon}\dx\dv\ds \leqs 2R\|\rho_{0}\|_{L^{1}_{x}}+R^{2}\|\Div u \|_{\LL{1}{1}}.$$

For each $w \in \R$, using the support property with a small regularization argument, we may use $v \to \sign(v-w)$ as a test function in to find the $\p$ almost sure equality $$\begin{aligned}
\label{eq:testForFixedw}
2\int_{0}^{T}\int_{\R^{d}}m_{\epsilon}(s,w,x)\dx\ds &= \iint_{\R^{d+1}}\chi(\rho_{0})\text{sign}(v-w)\dx\dv - \iint_{\R^{d+1}}f_{\epsilon}(T)\text{sign}(v-w)\dx\dv \nonumber \\
&-2w\int_{0}^{T}\int_{\R^{d}}\Div u_{\epsilon}(s,x)f_{\epsilon}(s,x,w)\dx\ds.\end{aligned}$$ Using that $|f_{\epsilon}| \leq 1$ by together with Lemma \[lem:unweightedL1bound\], equality implies that $\p$ almost surely $$\label{eq:testForFixedw2}
\int_{0}^{T}\int_{\R^{d}}m_{\epsilon}(s,w,x)\dx\ds \leqs  \|\rho_{0}\|_{L^{1}_{x}}+|w|\| \Div u \|_{L^{1}_{t,x}}.$$ Since both sides of are continuous in $w$ (by inspection of ), there is a single $\p$ null set where holds for all $w \in \R$. Hence, integrating both sides of yields .

In view of Hypothesis \[hyp:noiseFlux\], we may choose a $v_{0} \in \R$ such that $$\label{eq:defOfCutoffVelocity}
\inf_{|v| \geq v_{0}}b^{2}(v) \geq \frac{1}{2}\lambda.$$ The parameters $v_{0}$ and $\lambda$ will play a role in the following sub-solution Lemma.

\[lem:sub-Solution\] Assume that $u$ satisfies Hypothesis \[hyp:drift:subCritical\]. For each $p>0$, there exists an $\epsilon_{0}(p)$ such that for all $\epsilon<\epsilon_{0}$, there exists a function $\varphi_{\epsilon} \in L^{1}_{t}(W^{2,1}_{x})$ with $\partial_{t}\varphi_{\epsilon} \in L^{1}_{t,x}$ such that for almost all $(t,x) \in [0,T] \times \R^{d}$ and all $|v|\geq |v_{0}|$, $$\label{eq:subSolution} \partial_{t}\varphi_{\epsilon}+u_{\epsilon} \cdot \nabla_{x}\varphi_{\epsilon} + p(\Div_{x} u_{\epsilon})_{-} \varphi_{\epsilon} + \frac{1}{2}b^{2}(v)\Delta_{x} \varphi_{\epsilon}  \leq M ,$$ where $M$ 