A drift sublevel enlargement of a small set is petite

If $C=\mathrm{\varnothing }$, then $PV\le \lambda V$ everywhere; iterating gives $1\le P^{N}V(x)\le {\lambda }^{N}V(x)$ for every state $x$ and $N$, impossible for sufficiently large $N$ because $V(x)<\mathrm{\infty }$. Thus the hypotheses force $C\mathrm{\ne }\mathrm{\varnothing }$. Let ${\tau }_C=\inf \{n\ge 0:X_n\in C\}$. The killed kernel $K(x,A)=1_{C^c}(x)P(x,A)$ satisfies $KV\le \lambda V$ off $C$. Since $V\ge 1$, induction gives $K^{N}1(x)\le K^{N}V(x)\le {\lambda }^{N}V(x)$. Hence for $x\in \{V\le d\}$, ${\mathrm{Pr}}_x({\tau }_C\le N)\ge 1-d{\lambda }^N$; for $x\in C$, ${\tau }_C=0$.

For each $t\le N$, restrict paths to the disjoint event $\{{\tau }_C=t\}$ and apply [F5] followed by the small-set minorization. This gives $P^{t+m}(x,A)\ge ϵ{\mathrm{Pr}}_x({\tau }_C=t)\nu (A)\mathrm{.}$ Summing over $t$ yields ${\sum }_{t=0}^N{P}^{t+m}(x,A)\ge ϵ(1-d{\lambda }^N)\nu (A)$ uniformly on $D$.

Choose $N$ with $d{\lambda }^N\le 1\mathrm{/}2$ and assign weight $1\mathrm{/}(N+1)$ to times $m,\dots ,m+N$. The resulting probability mixture dominates $ϵ\nu \mathrm{/}[2(N+1)]$, so it satisfies the petite definition with positive multiplier $\alpha =ϵ\mathrm{/}[2(N+1)]$.