= Cutoff proof of invariance for a zero-flux diffusion density
{title2=$(\sigma^2p)'=2bp\quad\Longrightarrow\quad\int u(t)p=\int fp$}
If $q=\sigma^2p$ and $q'=2bp$, then $pLu=(qu_x)'/2$. Insert compact cutoffs $\chi_R$, integrate the backward equation in time, and integrate by parts twice to move spatial derivatives onto $\chi_R$. The error is bounded by $t\|u\|_\infty(CR^{-2}\|q\|_1+CR^{-1}\|q'\|_1)/2$. When these norms are finite it vanishes, proving $\int u(t)p=\int fp$ without assuming derivative behavior at infinity.
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