Given two Markov kernels on the same space $\mathfrak X$ and relative to the same dominating measure, $K_0(\cdot,\cdot)$ and $K_1(\cdot,\cdot)$, both ergodic with respective stationary distribution densities $\pi_0$ and $\pi_1$, is it always the case that the mixture kernel $$K_\alpha(\cdot,\cdot)=\alpha K_1(\cdot,\cdot)+(1-\alpha)K_0(\cdot,\cdot)$$ for $0<\alpha<1$, is also ergodic?
For instance, the result holds for generalised autoregressive processes (Brandt, 1986).