Immigration-death chain projected onto two states

The system has two channels, increments $+1$ and $-1$, and propensities $a_{+}(n)=1$ and $a_{-}(n)=n$; the death channel is inactive at $n=0$, so every positive-propensity poststate remains in ${\mathbb{N}}_0$ and the total rate $1+n$ is finite. Thus [F1] applies.

Let $V(n)=1+n$. Its sublevel set $\{n:V(n)\le m\}$ is finite for each integer $m\ge 1$, so $V$ is proper. For $n\ge 1$, $QV(n)=1(V(n+1)-V(n))+n(V(n-1)-V(n))=1-n\le 1+n=V(n)$; at $n=0$ the same value $QV(0)=1$ satisfies this bound. Hence [F2] with $c=1$ and [A1] proves the immigration-death chain is nonexplosive, as required by [F3].

On $F=\{0,1\}$, state $0$ jumps to $1$ at rate $1$; state $1$ jumps to $0$ at rate $1$ and exits to $2\mathrm{\notin }F$ at rate $1$. These off-diagonal rates are nonnegative, and the cemetery construction [F4] gives the displayed ${\tilde{Q}}_F$ with zero column sums. Multiplying by the lifted probability vector gives $p_0^{\mathrm{\prime }}=-p_0+p_1$, $p_1^{\mathrm{\prime }}=p_0-2p_1$, and $c_F^{\mathrm{\prime }}=p_1$, with initial vector $(1,0,0{)}^T$.

From $p_1^{\mathrm{\prime }}=p_0-2p_1$ we have $p_0=p_1^{\mathrm{\prime }}+2p_1$; differentiating and using $p_0^{\mathrm{\prime }}=-p_0+p_1$ yields $p_1^{\mathrm{\prime }\mathrm{\prime }}+3p_1^{\mathrm{\prime }}+p_1=0$, with $p_1(0)=0$ and $p_1^{\mathrm{\prime }}(0)=1$. The characteristic roots are $r_{\pm }=(-3\pm \sqrt{5})\mathrm{/}2$ and $r_{+}-r_{-}=\sqrt{5}$, so the unique solution is $p_1(t)=(e^{r_{+}t}-e^{r_{-}t})\mathrm{/}\sqrt{5}$.

Since $c_F^{\mathrm{\prime }}=p_1$ and $c_F(0)=0$, integrating the expression in step 2.1 gives the displayed formula for $c_F(t)$. By [F6], this coordinate is $\mathbb{P}({\tau }_F\le t)$ and equals the embedded killed-vector ${\mathrm{\ell }}^1$ error.

Fix $t>0$ and choose ordered times $0<s_1<s_2<s_3<t$. The joint density for jumps $0\to 1$ at $s_1$, $1\to 2$ at $s_2$, and $2\to 1$ at $s_3$, followed by no further jump through $t$, is $e^{-s_1}\cdot 1\cdot e^{-2(s_2-s_1)}\cdot 1\cdot e^{-3(s_3-s_2)}\cdot 2\cdot e^{-2(t-s_3)}$, which is strictly positive; integrating this positive density over the nonempty ordered-time simplex gives positive probability. Such paths contribute to the full marginal at state $1$ but have already been killed at their first exit from $F$.