The quasi-sure ratio ergodic theorem
Let \((P_{t})\) be the transition function of a right Markov process \((X_{t}, P_{x})\) and \(m\) be a conservative invariant measure. Taking a positive bounded function \(q\) with \(m(q)=1\), put \(\mu = q \cdot m\). The main purposes of this paper are to show the existence of the limit of \(\int^{t}_{0} P_{s} f(x) ds / \int^{t}_{0} P_{s} g(x) ds\) as \(t \rightarrow \infty\) except at most \(m\)-polar starting point \(x\) and to identify the limit with \(\mu(f/q|{\mathcal I})(x)/\mu(g/q|{\mathcal I})(x)\), where \({\mathcal I}\) is the \(\sigma\)-algebra of invariant sets. This refines the exceptional almost semipolar (or almost polar) set of symmetric Markov processes of \textit{M. Fukushima} [J. Math. Soc. Japan 26, 17-32 (1974; Zbl 0266.60057)] to almost polar set. For the proof, the author establishes a continuous version of the filling scheme. Let \(\overline{X}_{t}(w, r)=\left( X_{t}(w), r-t \right)\) be the backward space-time process and \(\overline{G}\) be the least \(\overline{X}\)-supermedian majorant of the \(\overline{X}\)-potential \(\overline{U}((g-f) \otimes 1)\). Then the pair of \(\overline{F} = \overline{G}-\overline{V} \left( (g - f) \otimes 1 \right)\) and \(\overline{H}=\overline{V} (g \otimes 1) - \overline{G}\) is called the filling scheme of \((f,g)\). Noting that the \(t\)-section \(U_{t} f\) of \(\overline{U} f \) can be written as \(\int^{t}_{0} P_{s} f(x)ds\) and writing \(U_{t}f / U_{t}g \) by means of the filling scheme, the proof is performed by reducing to the property of the set \(Z=\{x;F_\infty(x)=0\}\). The ratio limit theorems of positive additive functionals and resolvents are also shown as other applications of the filling scheme.
scientific article; zbMATH DE number 1164328