Equivalence of exponential ergodicity and \(L^ 2\)-exponential convergence for Markov chains.
Let \(Q\) be an irreducible regular \(Q\)-matrix on a countable state space \(E\). Assume that there exists an invariant probability measure \(\pi\) for the corresponding transition probability matrix \(P(t) = (p_ {ij}(t),\; i,j\in E)\). We say that \(P(t)\) is exponentially ergodic if there exists \(\alpha >0\) such that \(| p_ {ij}(t)-\pi _ {j}| \leq C_ {ij}\exp (-\alpha t)\), \(t\geq 0\), for every \(i,j\in E\) and a constant \(C_ {ij}\). This property is equivalent to exponential decay of \(\| p_ {i\bullet }-\pi \| _ {\text{var}}\) as \(t\to \infty \), where \(\| \cdot \| _ {\text{var}}\) denotes the total variation norm on the space of finite signed measures on \(E\). \(P(t)\) is said to be \(L^ 2\)-exponentially ergodic provided there exists \(\beta >0\) such that \(\| P(t)f - \pi (f)\| _ {L^ 2(\pi )} \leq \| f-\pi (f)\| _ {L^ 2(\pi )}\exp (-\beta t)\) for all \(f\in L^ 2(\pi )\) and \(t\geq 0\). It is shown that \(L^ 2\)-exponential ergodicity implies exponential ergodicity, and sufficient conditions are given for the converse implication to hold. In particular, it is so if \(P(t)\) is reversible with respect to \(\pi\). The last statement is then generalized to arbitrary state spaces: Let \((P_ {t})\) be a Markov semigroup on a measurable state space \((E,{\mathcal E})\), reversible with respect to a probability measure \(\pi \). Then \((P_ {t})\) is \(L^ 2\)-exponentially ergodic if and only if for every probability measure \(\mu \) absolutely continuous with respect to \(\pi\) and satisfying \(d\mu/d\pi \in L^ 2(\pi )\) there exists a constant \(C_ \mu \) such that \(\| P^ {*}_ {t}\mu - \pi \| _ {\text{var}} \leq C_ \mu \exp ( -\beta t)\), \(t\geq 0\). A discrete time version of this result has been proven recently by \textit{G. O. Roberts} and \textit{J. S. Rosenthal} [Electron. Commun. Probab. 2, 13--25 (1997; Zbl 0890.60061)].
scientific article; zbMATH DE number 2098079