## 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.

### MSC:

 60J45 Probabilistic potential theory 60J50 Boundary theory for Markov processes

Zbl 0754.00008
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