A Route to Harris ergodicity for Non-Feller Markov Kernels

For non-Feller Markov kernels satisfying a quasi-Feller factorization, the compact-petite-set criterion gives petite compact sets under $\psi$-irreducibility when the support of $\psi$ has non-empty interior. Thus, for coercive Lyapunov functions, the sublevel sets used in the Harris–Lyapunov argument are petite. Under aperiodicity they are small. The Hairer–Mattingly contraction theorem then yields geometric ergodicity in weighted total variation under the usual geometric drift condition.