 Solution

At leading order, ${\mathcal{L}}_0^{\ast }[p_0(y\mid x)p_0(x,t)]=0$. For fixed $x$, the conditional stationary law therefore satisfiesThis is an immigration--death process, whose stationary distribution is Poisson with meanThe mean is finite only for $x>0$. The assumption $X(0)\ne 0$ and the pair-coalescence propensity $x(x-1)$ ensure that the slow process cannot remove its last $X$ molecule, so the reduced model remains in that domain.