$$\label{eq:Def:ES:velocityDecay}
    \E \left ( \int_{0}^{T}\iint_{\R^{d+1}}(1+|v|^{p})|f|\dx\dv\dt \right )^{q} < \infty.$$

3.  There exists a kinetic measure $m$ such that for all $\varphi \in C^{1}([0,T];\, C^{\infty}_{c}(\R^{d+1}))$, a.e. in $\Omega \times [0,T]$, $$\begin{aligned}
    \label{eq:ES:Def:weakForm}
    \iint_{\R^{d+1}}f(t)\varphi \dx \dv &= \iint_{\R^{d+1}}\chi(\rho_{0})\varphi \dx \dv+ \int_{0}^{t}\iint_{\R^{d+1}}f[\partial_{t}\varphi+a \cdot \nabla_{x,v}\varphi+(1/2)b^{2}(v) \Delta \varphi]\dx \dv \ds \nonumber \\
    &+ \int_{0}^{t}\iint_{\R^{d+1}}f b\nabla_{x} \varphi \dx \dv \cdot \dW_{s}-\int_{0}^{T}\iint_{\R^{d+1}}1_{[0,t]}(s)\partial_{v}\varphi\dm .\end{aligned}$$

The formal a priori estimate {#subsec:aprioribds}
----------------------------

In this section, we give a formal argument which shows how to use the noise to avoid concentrations in the density. We work directly at the level of the “fluid” equation , rather than the kinetic formulation , where we will ultimately perform the rigorous analysis leading to Theorem \[thm:MainResult\]. For simplicity, we restrict our attention to non-negative solutions and focus only on obtaining an $L^{2}_{x}$ bound. Namely, our goal is to show that if $\rho_{0} \in L^{1}_{x} \cap L^{2}_{x}$, then $$\label{eq:formalBd:LpBound}
\rho \in L^{\infty}([0,T]; \, L^{2}(\Omega \times \R^{d})).$$ In the computations below, we pretend we are given a regular solution $\rho$ to and neglect the contribution of the entropy dissipation. $$\text{Step 1: Formal entropy balance}$$ Given an entropy $S:\R_{+} \to \R$, define a flux $\Phi: \R_{+} \to \R$ by $\Phi'(\rho)=S'(\rho)b(\rho)$. Since the noise is in Stratonovich 