the proof of the lemma using results from these papers while correcting for the differences in the systems, including changes in dimension and the appearance of a diffusion operator with variable coefficients. Consider the linear equation (\[LinVlasov\]) and define $$\mcL h:= \partial_t h + \hat{v}_1 \partial_x h + F \cdot \nabla_v h - \nabla\cdot(D \nabla_v h).$$ We first comment that solutions $h \in L^\infty((0,T); L^2(\mathbb{R}^3))$ of the equation $\mcL h = g$ exist for any $T > 0$ and $g \in L^\infty((0,T); L^\infty(\mathbb{R}^3))$, and this follows directly from either a variational argument [@Degond], the use of Green’s functions [@VictoryODwyer], or by properties of the heat equation on a Riemannian manifold [@CalogeroRVMFP]. With this, we prove a positivity result:

\[Lnew\] Let $T > 0$ be given. Assume $h_0 \in L^2(\bfR^3)$ and $g \in L^\infty((0,T); L^2(\bfR^3))$ are given with $h\in L^\infty((0,T); L^2(\bfR^3))$ satisfying $\mcL h = g \geq 0$ and $h(0,x,v) = h_0(x,v) \geq 0$. Then, $h(t,x,v) \geq 0$ for all $t \geq 0, x \in \bfR, v \in \bfR^2$.

Let $\lambda > \frac{1}{2}{\ensuremath{\left\| \nabla_v \cdot F \right\|}}_\infty$ be given. Define $u(t,x,v) = e^{-\lambda t} h(t,x,v)$ and $f(t,x,v) = e^{-\lambda t}  g(t,x,v)$. These functions then satisfy $$\label{lambda} 
\left \{ \begin{gathered}
\mcL   u 