to obtain the existence of a generalized solution to . Towards this end, it is convienient to re-write in the form $$\label{eq:BGK2}
\begin{cases}
\partial_{t}\fe + \Div_{x,v}(a_{\epsilon}f_{\epsilon}) +\Div_{x}(bf_{\epsilon} \circ\dot{W})=\partial_{v}m_{\epsilon}\\
f_{\epsilon} \mid_{t=0}=\chi(\rho_{0}), 
\end{cases}$$ where $$\label{eq:BGKKineticMeasure}
m_{\epsilon}(t,x,v,\omega):=\frac{1}{\epsilon}\int_{-\infty}^{v}\left [\chi(w,\rho_{\epsilon}(t,x,\omega))-f_{\epsilon}(t,x,w,\omega)\right]\dee w.$$ Note that the support property ensures that $m_{\epsilon}$ is well defined. Moreover, it can be shown that the sign property of $f_{\epsilon}$ ensures that $m_{\epsilon}$ is non-negative (see [@MR2064166], Theorem 2.2.1). Our goal now is to obtain a priori bounds on $\{f_{\epsilon}\}_{\epsilon>0}$ and $\{m_{\epsilon}\}_{\epsilon>0}$ and to introduce a notion of generalized kinetic solution to describe the equation satisfied by their weak limit points.

A priori estimates {#subssec:trueApriori}
------------------

\[lem:unweightedL1bound\] For each $\epsilon>0$, the following inequality holds $\p$ almost surely $$\label{eq:unweightedL1bound}
\sup_{t\in [0,T]}\iint_{\R^{d+1}}|f_{\epsilon}(t,x,v)|\dx \dv \leq \|\rho_{0}\|_{L^{1}_{x}}.$$

Using the support property of $f_{\epsilon}$ and a small regularization argument, we may use $v \to \sign(v)$ as a test function in the weak formulation to find: for each $t \in [0,T]$, it holds $\p$ almost surely that $$\label{eq:L1Balance}
\iint_{\R^{d+1}}|f_{\epsilon}(t)|\dx\dv + 2\int_{0}^{t}\int_{\R^{d}}m_{\epsilon}(x,0)\dx\ds = \iint_{\R^{d+1}}\sign(v)\chi(\rho_{0})\dx\dv.$$ Note that we have used the sign property to write $\sign(v)f_{\epsilon}=|f_{\epsilon}|$. Since $m_{\epsilon}$ is non-negative and $\sign(v)\chi(\rho_{0})=|\chi(\rho_{0})|$, the inequality implies .

\[lem:LowVelBdsonKM\] For each $\epsilon>0$ and each $R>0$, the following inequality holds $\p$ almost surely 