	\begin{align*}
		\begin{cases} \partial_t f_{\geq N+1} +v\cdot \nabla f_{\geq N+1} = -\sigma f_{\geq N+1} + \mathcal{L}{f_{ N}}+ \mathcal{L}{f_{\geq N+1}} \,,\\
			f_{\geq N+1}(x,t=0,v) = 0\,.
		\end{cases}
	\end{align*}
