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**Some applications of quasi-boundedness for excessive measures.**
*(English)*
Zbl 0768.60064

Séminaire de probabilités XXVI, Lect. Notes Math. 1526, 485-497 (1992).

[For the entire collection see Zbl 0754.00008.]

Let \(\xi\) and \(m\) be excessive measures for a right Markov process \(X=(X_ t,P_ x)\). A stationary Markov process \((Y_ t,Q_ \xi)\), which has random birth time \(\alpha\), death time \(\beta\), the same transition function as \(X\) and \(Q_ \xi(Y_ t\in\cdot)=\xi_ 1\), is associated with \(\xi\). \(Q_ \xi\) is called the Kuznetsov measure associated with \(\xi\) and \(X\). \(\xi\) is said quasi-bounded by \(m\) \((\xi\in Qbd(m)\) in notation) if \(\xi=\sum_ k\xi_ k\) for some excessive measures \(\xi_ k\) such that \(\xi_ k\leq m\). Let \(\xi=\nu V+\eta\) and \(m=\mu V+\gamma\) be the Riesz decomposition of excessive measures \(\xi\) and \(m\) into the sum of potential part and harmonic part. It is proved that \(Q_ \xi\ll Q_ m\) if and only if \(\eta\in Qbd(\gamma)\) and \(\nu\ll\mu\). Moreover, when \(Q_ \xi\ll Q_ m\), it is shown that the density \(J=dQ_ \xi/dQ_ m\) is given by \(J=\lim_{t\downarrow\alpha}u(Y_ t)\) on \(W^ c_ p\) and \(J=f(Y_{\alpha+})\) on \(W_ p\), where \(u\) and \(f\) are suitable versions of \(d\eta/d\gamma\) and \(d\nu/d\mu\), respectively, and \(W_ p\) is a set of paths such that \(Q_{\mu V}=Q_ m(\cdot;W_ p)\), \(Q_ \gamma=Q_ m(\cdot;W^ c_ p)\). A view connecting this result with a Fatou-type boundary limit theorem is also stated. Finally a general form of Kuran’s theorem which characterizes the regularity of a point \(x\in D^ c\) by the quasi-boundedness of \(\varepsilon_ xU|_ D\) relative to \(m|_ D\) is given.

Let \(\xi\) and \(m\) be excessive measures for a right Markov process \(X=(X_ t,P_ x)\). A stationary Markov process \((Y_ t,Q_ \xi)\), which has random birth time \(\alpha\), death time \(\beta\), the same transition function as \(X\) and \(Q_ \xi(Y_ t\in\cdot)=\xi_ 1\), is associated with \(\xi\). \(Q_ \xi\) is called the Kuznetsov measure associated with \(\xi\) and \(X\). \(\xi\) is said quasi-bounded by \(m\) \((\xi\in Qbd(m)\) in notation) if \(\xi=\sum_ k\xi_ k\) for some excessive measures \(\xi_ k\) such that \(\xi_ k\leq m\). Let \(\xi=\nu V+\eta\) and \(m=\mu V+\gamma\) be the Riesz decomposition of excessive measures \(\xi\) and \(m\) into the sum of potential part and harmonic part. It is proved that \(Q_ \xi\ll Q_ m\) if and only if \(\eta\in Qbd(\gamma)\) and \(\nu\ll\mu\). Moreover, when \(Q_ \xi\ll Q_ m\), it is shown that the density \(J=dQ_ \xi/dQ_ m\) is given by \(J=\lim_{t\downarrow\alpha}u(Y_ t)\) on \(W^ c_ p\) and \(J=f(Y_{\alpha+})\) on \(W_ p\), where \(u\) and \(f\) are suitable versions of \(d\eta/d\gamma\) and \(d\nu/d\mu\), respectively, and \(W_ p\) is a set of paths such that \(Q_{\mu V}=Q_ m(\cdot;W_ p)\), \(Q_ \gamma=Q_ m(\cdot;W^ c_ p)\). A view connecting this result with a Fatou-type boundary limit theorem is also stated. Finally a general form of Kuran’s theorem which characterizes the regularity of a point \(x\in D^ c\) by the quasi-boundedness of \(\varepsilon_ xU|_ D\) relative to \(m|_ D\) is given.

Reviewer: Y.Oshima (Kumamoto)