0}^{t+\epsilon}(u(s,\cdot)*\eta_{\epsilon})(x)\ds, \quad \quad a_{\epsilon}(x,v):=[u_{\epsilon}(t,x),-v\Div u_{\epsilon}(t,x)].$$ We now define a notion of weak solution to .

\[Def:BGKWeakSol\] A measurable mapping $f_{\epsilon}:\Omega \times [0,T] \times \R^{d+1} \to \R$ is called a weak solution to starting from $\chi(\rho_{0})$ provided that

1.  \[item:BGKDef:ContandMeas\] The mapping $f_{\epsilon}: \Omega \times [0,T] \to L^{\infty}_{x,v}$ is weak-$*$ predictable. Moreover, sample paths of $f_{\epsilon}$ belong to $C([0,T]; L^{1}_{x,v})$ with probability one.

2.  \[item:BGKDef:equation\] For all $\varphi \in C^{1}([0,T];C^{\infty}_{c}(\R^{d+1}))$ and all $t \in [0,T]$ it holds $\p$ almost surely that $$\begin{aligned}
    \label{eq:BGK:Def:weakForm}
    \iint_{\R^{d+1}}f_{\epsilon}(t)\varphi(t) \dx \dv &= \iint_{\R^{d+1}}\chi(\rho_{0})\varphi(0) \dx \dv\nonumber\\
    &+ \int_{0}^{t}\iint_{\R^{d+1}}f_{\epsilon}[\partial_{t}\varphi+a_{\epsilon} \cdot \nabla_{x,v}\varphi+(1/2)b^{2}\Delta\varphi]\dx \dv \ds \nonumber \\
    &+ \int_{0}^{t}\iint_{\R^{d+1}}f_{\epsilon} b\nabla_{x} \varphi \dx \dv \cdot \dW_{s}+\epsilon^{-1}\int_{0}^{t}\iint_{\R^{d+1}}\left [\chi(\rho_{\epsilon})-f_{\epsilon} \right]\varphi\dx\dv\ds,\end{aligned}$$ where $$\label{eq:BGKDef:rho}
    \rho_{\epsilon}=\int_{\R}f_{\epsilon}\dv.$$

3.  It holds a.e. in $\Omega \times [0,T] \times \R^{d+1}$ that $$\begin{aligned}
    \label{BGK:signProp} \sign(v)f_{\epsilon}(\omega,t,x,v)&=|f_{\epsilon}(\omega,t,x,v)|\leq 1.\\
    \label{BGK:suppProp}f_{\epsilon}(\omega,t,x,v)&=0 \quad \text{if} \quad |v| \geq \|\rho_{0}\|_{L^{\infty}_{x}}\exp(t\|\Div u_{\epsilon}\|_{L^{\infty}_{t,x}}).\end{aligned}$$

The following result is proved in the appendix by a classical fixed point argument.

\[prop:BGKExistence\] For each $\rho_{0} \in L^{1}_{x} \cap L^{\infty}_{x}$ and each $\epsilon>0$, there exists a weak solution $f_{\epsilon}$ to starting from $\chi(\rho_{0})$.\

By Proposition \[prop:BGKExistence\], for each $\epsilon>0$ we may construct a weak solution $f_{\epsilon}$ to , yielding a sequence $\{f_{\epsilon}\}_{\epsilon>0}$. In this section, we study the hydrodynamic limit $\epsilon \to 0$ in order 