Killed finite-state projection and cemetery lift

Let $X$ be a given nonexplosive chain as in Given countable nonexplosive pure-jump chain with finite rates, with initial law $p_0$ on $S$, and let $F\subseteq S$ be finite and nonempty. Its first exit time from $F$ is

${\tau }_F:=\inf \{s\ge 0:X_s\mathrm{\notin }F\},$

with ${\tau }_F=0$ when $X_0\mathrm{\notin }F$. Adjoin a new state $†\mathrm{\notin }S$ and define the killed projection $Y_t=X_t$ for $t<{\tau }_F$ and $Y_t=†$ for $t\ge {\tau }_F$. The killed mass vector is $u_F(t,x):=\mathbb{P}(Y_t=x)$ for $x\in F$, and the cemetery mass is $c_F(t):=\mathbb{P}(Y_t=†)$. Initially, $u_F(0,x)=p_0(x)$ for $x\in F$ and $c_F(0)=p_0(S\setminus F)$.

For $x\in F$ let $\lambda (x)={\sum }_{y\mathrm{\ne }x}q(x,y)$. The killed principal subgenerator on $F$ has entries $(Q_F{)}_{yx}=q(x,y)$ for $y\mathrm{\ne }x$ in $F$ and $(Q_F{)}_{xx}=-\lambda (x)$; its column sum is $-{\sum }_{y\mathrm{\notin }F}q(x,y)$. The cemetery lift is the finite matrix ${\tilde{Q}}_F$ on $F\cup \{†\}$ with these same entries on $F$, with

${\tilde{Q}}_{†x}=\lambda (x)-{\sum }_{\begin{array}{c}y\in F\\ y\mathrm{\ne }x\end{array}}q(x,y)={\sum }_{y\mathrm{\notin }F}q(x,y)(x\in F),{\tilde{Q}}_{x†}=0(x\in F),{\tilde{Q}}_{††}=0.$

Every column of ${\tilde{Q}}_F$ sums to zero and its off-diagonal entries are nonnegative, so it is a finite column generator in the sense of Finite column generator. The cemetery state is absorbing; trajectories that return to $F$ after their first exit remain at $†$ in this projection.