= Poisson Stein equation
{c}
{title2=$\mu f(k+1)-kf(k)=g(k)-\mathbb Eg(Z),\quad Z\sim\operatorname{Poisson}(\mu)$}
For indicator test functions $g$, the equation has a solution with $\|\Delta f\|_\infty\le1$. One proof uses an <immigration--death process> with immigration rate $\mu$ and per-particle death rate one. Its potential $u(k)=-\int_0^\infty(\mathbb Eg(Z_t^k)-\mathbb Eg(Z))\,dt$ satisfies its generator equation, and $f(k)=u(k)-u(k-1)$. Coupling two additional initial particles bounds the second difference of $u$ by $\int_0^\infty2e^{-2t}dt=1$.
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