Cutoff proof of invariance for a zero-flux diffusion density ($({\sigma }^{2}p{)}^{\prime }=2bp⟹\int u(t)p=\int fp$)

If $q={\sigma }^{2}p$ and $q^{\prime }=2bp$, then $pLu=(q{u}_x{)}^{\prime }/2$. 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 $t\Vert u{\Vert }_{\infty }(C{R}^{-2}\Vert q{\Vert }_1+C{R}^{-1}\Vert q^{\prime }{\Vert }_1)/2$. When these norms are finite it vanishes, proving $\int u(t)p=\int fp$ without assuming derivative behavior at infinity.