Poisson Stein equation ($\mu f(k+1)-kf(k)=g(k)-\mathbb{E}g(Z),Z\sim \mathrm{Poisson}(\mu )$)

For indicator test functions $g$, the equation has a solution with $\Vert \Delta f{\Vert }_{\infty }\le 1$. One proof uses an immigration--death process with immigration rate $\mu$ and per-particle death rate one. Its potential $u(k)=-{\int }_0^{\infty }(\mathbb{E}g(Z_t^k)-\mathbb{E}g(Z))dt$ satisfies its generator equation, and $f(k)=u(k)-u(k-1)$. Coupling two additional initial particles bounds the second difference of $u$ by ${\int }_0^{\infty }2e^{-2t}dt=1$.