Small-set minorization and geometric drift

For a probability kernel $P$ on $(\mathsf{X},\mathcal{X})$ (Measure kernel and probability kernel), a measurable set $C$ is $(m,ϵ,\nu )$-small if $m\ge 1$ is an integer, $0<ϵ\le 1$, $\nu$ is a probability measure (Probability measures and probability spaces), and $P^m(x,A)\ge ϵ\nu (A)(x\in C,A\in \mathcal{X})\mathrm{.}$ The inequality is pointwise in $x$ and holds for every measurable $A$.

A geometric drift inequality outside $C$ consists of a finite-valued measurable $V:\mathsf{X}\to [1,\mathrm{\infty })$, constants $0<\lambda <1$ and $0\le b<\mathrm{\infty }$, and $PV(x):=\int V(y)P(x,dy)\le \lambda V(x)+b{1}_C(x)(x\in \mathsf{X})\mathrm{.}$ The terms small set and drift are separate hypotheses; neither definition asserts the existence of the other.

When an invariant probability $\pi$ is specified (Invariant and stationary distribution for a Markov kernel), an extended geometric drift certificate finite $\pi$-almost everywhere allows measurable $V:\mathsf{X}\to [1,+\mathrm{\infty }]$ with $\pi (V<\mathrm{\infty })=1$ and the same pointwise inequality, interpreted in the nonnegative extended reals. This is a separate convention from the finite-valued drift function above. The geometric theorem treats this certificate on $E=\{V<\mathrm{\infty }\}$; the original finite-valued convention gives the corresponding bound on the whole state space.