Hypothesis \[hyp:drift:subCritical\]. For each $p \geq 1$ and $\epsilon<\epsilon_{0}(p)$, it holds that for all $q\geq 2$ $$\label{eq:AveragedVelocityDecay}
 \E \left ( \sup_{s \in [0,T]}\iint_{\R^{d+1}}|v|^{p}|f_{\epsilon}(s)|\dx\dv \right )^{q}+\E\left ( \int_{0}^{T}\iint_{\R^{d+1}}|v|^{p-1}m_{\epsilon}\dx\dv\ds \right )^{q} \leq C\big [ 1+\|\rho_{0}\|_{L^{p+1}_{x}}^{(p+1)q} \big ],$$ where $C$ depends on $q$, $p$, $T$, $b$ and $u$.\

The proof proceeds similarly to the arguments presented in Section \[subsec:aprioribds\], combining Lemma \[lem:sub-Solution\] to obtain a suitable weight, together with Gronwall’s inequality.\
*Step 1: Application of Lemma \[lem:sub-Solution\]*\
The main inequality leading to involves the sub-solution $\varphi_{\epsilon}$ constructed in Lemma \[lem:sub-Solution\]. We now claim that $\p$ almost surely it holds for each $t \in [0,T]$ $$\begin{aligned}
\label{eq:Energy Balance}
\iint_{\R^{d+1}}&|v|^{p}|f_{\epsilon}(t)|\dx\dv+p\int_{0}^{t}\iint_{\R^{d+1}}|v|^{p-1} m_{\epsilon} \dx \dv \ds \nonumber  \\
&\leq \frac{1}{\min\{1,\frac{1}{2p}\}} \left [ \frac{M}{p+1}\|\rho_{0}\|_{L^{p+1}}^{p+1}+M\int_{0}^{t}\iint_{\R^{d+1}}|v|^{p}|f_{\epsilon}(s)|\dx\dv\ds+I_{\epsilon,1}(t) +I_{\epsilon,2}(t)\right ],\end{aligned}$$ where $$\begin{aligned}
&I_{\epsilon,1}(t)=\int_{0}^{t}\iint_{\R^{d+1}}1_{\{|v|<|v_{0}|\}}|v|^{p}|f_{\epsilon}|[\partial_{t}\varphi_{\epsilon}+u_{\epsilon} \cdot \nabla_{x}\varphi_{\epsilon}+p(\Div u_{\epsilon})_{-}\varphi_{\epsilon}+(1/2)b^{2}(v)\Delta_{x} \varphi_{\epsilon}]\dx \dv \ds\\
&I_{\epsilon,2}(t)=\int_{0}^{t}\iint_{\R^{d+1}}|v|^{p}|f_{\epsilon}| b\nabla_{x} \varphi_{\epsilon} \dx \dv \cdot \dW_{s}.\end{aligned}$$ To establish , the support property combined with a small regularization argument allows to take $(t,x,v) \to \sign(v)|v|^{p}\varphi_{\epsilon}(t,x)$ as a test function in the weak formulation to obtain: $$\begin{aligned}
\label{eq:aPriori:MainIdentityPreExp}
\iint_{\R^{d+1}}&|v|^{p}|f_{\epsilon}(t)|\varphi_{\epsilon}(t) \dx \dv+p\int_{0}^{t}\iint_{\R^{d+1}}\varphi_{\epsilon}|v|^{p-1}m_{\epsilon}\dx\dv\ds = \iint_{\R^{d+1}}|v|^{p}|\chi(\rho_{0})|\varphi_{\epsilon}(0) \dx \dv \nonumber\\
+&\int_{0}^{t}\iint_{\R^{d+1}}|v|^{p}|f_{\epsilon}|[\partial_{t}\varphi_{\epsilon}+u_{\epsilon} \cdot \nabla_{x}\varphi_{\epsilon}+p\Div u_{\epsilon}\varphi_{\epsilon}+(1/2)b^{2}(v)\Delta_{x} \varphi_{\epsilon}]\dx \dv \ds \nonumber \\
+&\int_{0}^{t}\iint_{\R^{d+1}}|v|^{p}|f_{\epsilon}| b\nabla_{x} \varphi_{\epsilon} \dx \dv \cdot \dW_{s},\nonumber  \end{aligned}$$ where we have split $a_{\epsilon}$ into $x$ and $v$ components and 