Right dual process for semidynamical systems
The paper continues the author's analysis of semidynamical systems (SDS) and related structures, see, e.g., \textit{M. Bezzarga} [Rev. Roum. Math. Pures Appl. 42, No. 1--2, 15--30 (1997; Zbl 1071.31501)]. In loose terms, an SDS is a measurable space \((X,\mathcal{B})\) with an action of a 1-parameter semigroup \(\Phi\) of measurable maps \((\Phi: \mathbb{R}_+\times X\to X\)) with a fixed point \(\omega\), capturing the \(\Phi\)-trajectory of any \(x \in X_0 = X \setminus \{\omega\}\) at the time \(\rho(x) \in \overline{\mathbb{R}}_+\). ``Transient'' SDS endows \(X_0\) with order, measure \(\Lambda\), suitable topology \(\mathcal{T}^0_{\Phi}\) and metrics. Resolvents \(\mathcal{V}\) and \(\mathcal{V}^*\) of kernels arise on \(X_0\); their excessive functions are characterized. \(\mathcal{V}\) corresponds to the ``deterministic'' semigroup \(\mathbb{H}\) of Koopman operators induced by \(\Phi\); its ``dual'' \(\mathcal{V}^*\) proves to be a resolvent of the constructed ``dual'' semigroup \(\mathbb{P} = (P_t)_{t\geq0}\) of kernels s.t. \(\int_{X_0} g(P_tf)d\Lambda = \int_{X_0} f(H_tg)d\Lambda\) for measurable positive \(f,g\). \(\mathbb{P}\) is sub-Markovian iff ``maximal'' trajectories are disjoint. The condition imposed implies that connected components of \((X_0,\mathcal{T}^0_{\Phi})\) are unions, at most countable, of maximal trajectories. A so-called Doob \(u\)-transformation produces from \(\mathbb{P}\) on connected \(X_0\) a sub-Markovian semigroup \(\mathbb{P}^u\). Invariant probabilities on \(X_0\) w.r.t. \(\mathbb{P}^u\) \((\mathbb{H}\) has none) are mixtures of atoms at starting points of maximal trajectories. The right ``dual'' process, constructed on the space of ``cotrajectories'' for SDS, is a Markov process with transition semigroup \(\mathbb{P}^u\).
scientific article; zbMATH DE number 2093724