Geometric drift yields a bounded small set and bivariate drift

Choose ${\lambda }^{\mathrm{\prime }}\in (\lambda ,1)$ and then $d_0>b\mathrm{/}({\lambda }^{\mathrm{\prime }}-\lambda )$ large enough that $C_0=C\cap \{V\le d_0\}$ has positive $\varphi$-measure; such a choice exists since $V$ is finite and $C$ is positive. The set $C_0$ remains $(m,ϵ,\nu )$-small and is bounded in $V$. For $x\in C\setminus C_0$, $b\le ({\lambda }^{\mathrm{\prime }}-\lambda )V(x)$, while outside $C$ the original drift has no $b$ term. Hence $PV\le {\lambda }^{\mathrm{\prime }}V+b{1}_{C_0}$.

By [F3], $\pi (C_0)>0$. Integrating the minorization over $C_0$ gives $\nu \ll \pi$. Iteration of the original drift yields $P^{m}V(x)\le {\lambda }^{m}V(x)+b\mathrm{/}(1-\lambda )$; for $x\in C_0$ the right side is finite and uniformly bounded. Since $P^m(x,\cdot )\ge ϵ\nu$, it follows that $\nu V\le {ϵ}^{-1}{P}^{m}V(x)<\mathrm{\infty }$. Irreducibility and $\varphi (C_0)>0$ give some $k\ge 0$ with $\nu P^k(C_0)>0$. Put ${\nu }^{\mathrm{\prime }}=\nu P^k$. For any $A$ with $\pi (A)=0$, invariance gives $\int P^k(x,A)\pi (dx)=\pi (A)=0$, so $P^k(x,A)=0$ for $\pi$-almost every $x$; since $\nu \ll \pi$, this implies ${\nu }^{\mathrm{\prime }}(A)=\int P^k(x,A)\nu (dx)=0$. Thus ${\nu }^{\mathrm{\prime }}\ll \pi$. Also ${\nu }^{\mathrm{\prime }}V<\mathrm{\infty }$, and $C_0$ is small at time $m+k$ with minorizer ${\nu }^{\mathrm{\prime }}$.

Choose any finite $d_1$ large enough that $2b\mathrm{/}((1-{\lambda }^{\mathrm{\prime }})(d_1+1))<1$. Put $D=C_0\cup \{V\le d_1\}$, which is bounded in $V$. By [F4], there are weights $(a_n)$, $\alpha >0$, and a probability $\mu$ such that ${\sum }_{n\ge 1}{a}_n{P}^n(x,\cdot )\ge \alpha \mu$ on $D$. Put $m_0=m+k$, so $C_0$ is $(m_0,ϵ,{\nu }^{\mathrm{\prime }})$-small. By [F1], every state reaches $C_0$ at some positive time; a countable union therefore gives $j\ge 1$ with $\mu P^j(C_0)>0$. Composition yields $\mu P^{j+m_0}\ge ϵ\mu P^j(C_0){\nu }^{\mathrm{\prime }}$. Thus the shifted mixture ${\sum }_{n\ge 1}{a}_n{P}^{n+j+m_0}$ makes $D$ petite with minorizer ${\nu }^{\mathrm{\prime }}$. Since ${\nu }^{\mathrm{\prime }}(C_0)>0$ and ${\nu }^{\mathrm{\prime }}\ll \pi$, [F5] gives some $\mathrm{\ell }\ge 1$, $\beta >0$, and probability $\eta$ with $P^{\mathrm{\ell }}(x,\cdot )\ge \beta \eta$ on $D$. Again [F1] gives $j^{\mathrm{\prime }}\ge 1$ with $\eta P^{j^{\mathrm{\prime }}}(C_0)>0$. Hence, for $L_0=\mathrm{\ell }+j^{\mathrm{\prime }}+m_0$, $P^{L_0}(x,\cdot )\ge {\delta }_0{\nu }^{\mathrm{\prime }}$, where ${\delta }_0=\beta ϵ\eta P^{j^{\mathrm{\prime }}}(C_0)>0$ on $D$. Each state also reaches a positive minorization into ${\nu }^{\mathrm{\prime }}$ by reaching $C_0$ and then running $m_0$ steps. For every sufficiently large $t$, [F6] gives ${\nu }^{\mathrm{\prime }}{P}^t\ge c_t{\nu }^{\mathrm{\prime }}$; hence $P^{L_0+t}(x,\cdot )\ge {\delta }_0{c}_t{\nu }^{\mathrm{\prime }}$ for $x\in D$.

Iterating the improved drift gives $P^{L}V\le {\lambda }^{\mathrm{\prime }L}V+B_0$ with $B_0=b\mathrm{/}(1-{\lambda }^{\mathrm{\prime }})$. Choose $t$ in step 3.1 large enough that $L=L_0+t$ also satisfies $a:={\lambda }^{\mathrm{\prime }L}+\frac{2B_0}{d_1+1}<1.$ If $(x,y)\mathrm{\notin }{D}^2$, at least one coordinate has $V>d_1$, so $h\ge (d_1+1)\mathrm{/}2$. Therefore $Q\otimes Qh\le {\lambda }^{\mathrm{\prime }L}h+B_0\le ah\mathrm{.}$ The set $D$ is bounded in $V$, and $QV$ is uniformly bounded on $D$ by the drift bound. The minorizer ${\nu }^{\mathrm{\prime }}$ has finite $V$-moment by step 2.1, so the residual laws have uniformly finite $V$-moment and their product expectation of $h$ is bounded.