Verifiable Conditions for Irreducibility, Aperiodicity and T-chain Property of a General Markov Chain

We consider in this paper Markov chains on a state space being an open subset of $\mathbb{R}^n$ that obey the following general non linear state space model: $p_{t+1} = F(p_t, \alpha(p_t,U_{t+1})), t \in \mathbb{N},$ where $(U_t)_{t \in \mathbb{N}^*}$ (each $U_t \in \mathbb{R}^p$) are i.i.d. random vectors, the function $\alpha$, taking values in $\mathbb{R}^m$, is a measurable typically discontinuous function and $(x,w) \mapsto F(x,w)$ is a $C^1$ function. In the spirit of the results presented in the chapter~7 of the Meyn and Tweedie book on "Markov Chains and Stochastic Stability", we use the underlying deterministic control model to provide sufficient conditions that imply that the chain is a $\varphi$-irreducible, aperiodic T-chain with the support of the maximal irreducibility measure that has a non empty interior. Using previous results on our modelling would require that the overall update function $(x,u) \mapsto F(x,\alpha(x,u) )$ is $C^\infty$ and that $U_1$ admits a lower semi-continuous density. In contrast, we assume that the function $(x,w) \mapsto F(x,w)$ is $C^1$, and that for all $x$, $\alpha(x,U_1)$ admits a density $p_x$ such that the function $(x,w) \mapsto p_x(w)$ is lower semi-continuous. Hence the function $(x,u) \mapsto F(x,\alpha(x,u) )$ may have discontinuities, contained within the function $\alpha$. We introduce the notion of a strongly globally attracting state and we prove that if there exists a strongly globally attracting state and a time step $k$, such that we find a $k$-path such that the $k^{\rm th}$ transition function starting from $x^*$, $F^k(x^*,.)$, is a submersion at this $k$-path, the the chain is a $\varphi$-irreducible, aperiodic, $T$-chain. We present two applications of our results to Markov chains arising in the context of adaptive stochastic search algorithms to optimize continuous functions in a black-box scenario.