notion of solution to . We introduce the Maxwellian $\chi: \R^{2} \to \R$ defined by $$\chi(\rho,v)=1_{(-\infty,\rho)}(v)-1_{(-\infty,0)}(v).$$ The kinetic formulation of is to ask that the kinetic density $f(t,x,v)=\chi(\rho(t,x),v)$ satisfies the SPDE

$$\label{eq:SPDE:kinetic}
\begin{cases}
\partial_{t}f +\Div_{x,v}\left (af\right )+\Div_{x}(b(v)f\circ \dot{W})=\partial_{v}m \quad \text{on} \quad [0,T] \times \R^{d}\\
f \mid_{t=0}=\chi(\rho_{0}),
\end{cases}$$

where $a(t,x,v)=[u(t,x),-v\Div u(t,x)]$ and $m: \Omega \to \mathcal{M}_{t,x,v}^{+}$ is a random variable on the space of non-negative, bounded measures on $[0,T] \times \R^{d+1}$. More specifically, we ask that $m$ be a kinetic measure in the following sense.

\[Def:Kinetic\_Measure\] A weak-$*$ measurable mapping $m: \Omega \to \mathcal{M}_{t,x,v}^{+}$ is called a kinetic measure provided that:

1.  For all $p \geq 0$ and $q \geq 1$ it holds $$\label{eq:Def:KMDefVelocityDecay}
    \E \left ( \int_{0}^{T}\iint_{\R^{d+1}}|v|^{p}\dee m \right )^{q}<\infty.$$

2.  For all $\varphi \in C^{\infty}_{c}(\R^{d+1})$, the mapping $$\label{eq:Def:KM:predictable}
    (\omega, t) \to \int_{0}^{T}\iint_{\R^{d+1}}1_{[0,t]}(s)\varphi(x,v)\dee m(\omega)$$ is predictable.

\[Def:Entropy\_Solution\] Given $\rho_{0} \in L^{1}_{x} \cap L^{\infty}_{x}$, a measurable mapping $\rho: \Omega \times [0,T] \times \R^{d} \to \R$ is called a kinetic solution to starting from $\rho_{0}$ provided the mapping $$(\omega,t,x,v) \to f(\omega,t,x,v):=\chi ( \rho(\omega,t,x),v)$$ has the following properties:

1.  \[ES:Def:item:Measurability\] The mapping $f:\Omega \times [0,T] \to L^{\infty}(\R^{d+1})$ is weak-$*$ predictable.

2.  For all $p \geq 0$ and $q \geq 1$ it holds that 