A note on capacitary measures of semipolar sets. (English) Zbl 0669.60070

The author considers a pair of standard processes with stochastic resolvents in strong duality, and he shows that for a Borel set S in the state space satisfying \(S\subset \{P_ S1<\epsilon\), \(\hat P_ S1<\epsilon \}\) for some \(\epsilon\in (0,1]\) (where \(P_ S\), \(\hat P_ S\) denote the respective hitting kernels) there exists a \(\sigma\)-finite measure \(\pi\) on S satisfying the following condition:
A Borel subset B of S is polar if and only if \(\pi (B)=0.\)
According to a more general result by C. Dellacherie, D. Feyel and G.Mokobodzki [Séminaire de probabilités XVI, Univ. Strasbourg 1980/81, Lect. Notes Math. 920, 8-28 (1982; Zbl 0496.60076)] this property characterizes S to be semipolar. The author shows, moreover, that the measure \(\pi\) may be chosen as the capacitary measure with respect to a \(\lambda\)-subprocess \((\lambda >0)\) provided \(\epsilon <1.\)
In a recent preprint “Remarks on a paper of Kanda”, P. J. Fitzsimmons has strengthened the author’s result and simplified its proof.
Reviewer: J.Steffens


60J45 Probabilistic potential theory
60J40 Right processes


Zbl 0496.60076
Full Text: DOI