Nonnegative kernel action and finite drift

Let $p$ be a transition matrix on countable $E$ and let $\varphi :E\to [0,+\mathrm{\infty }]$. Define the nonnegative kernel action

$P\varphi (x):={\sum }_{y\in E:p(x,y)>0}p(x,y)\varphi (y)\in [0,+\mathrm{\infty }],$

using the extended nonnegative sum of Series in the nonnegative extended real line. Terms with zero transition weight are omitted: The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, its order, and the arithmetic that is left undefined leaves $0\cdot (+\mathrm{\infty })$ undefined, while each displayed product has positive finite first factor and is defined even when $\varphi (y)=+\mathrm{\infty }$. If $\varphi$ is finite-valued and $P\varphi (x)<\mathrm{\infty }$, its drift is the finite real number

$L\varphi (x):=P\varphi (x)-\varphi (x)\mathrm{.}$

This agrees with the published discrete generator Discrete generator of a countable-state transition matrix on bounded functions. For an extended-valued $u$, write a first-step relation as $u=Pu+c$ in extended nonnegative arithmetic; do not define a drift by subtracting $+\mathrm{\infty }$ from $+\mathrm{\infty }$.