Summary
We show at first that if (P t) is a special standard semi-group on a locally compact space E with a countable base and if u is an excessive function, then the semi-group \(\left( {\tilde P_t^{\left( u \right)} } \right)\), superharmonic transform of (P t) by u, is special standard if (and only if) the function u is regular.
Then we prove that if two standard semi-groups in duality verifying Kunita-Watanabe's hypotheses are given and if g is a coexcessive function (excessive for the dual semi-group) then: 1) g is Borel measurable, 2) g is a. s. left continuous and has right limits on the sample path of the process \(\left( {X_{t^ - } } \right)_{0 < t < \zeta ,}\) 3) g has a.s. right and left limits on the sample path of the process \(\left( {X_t } \right)_{0 < t < \zeta }\) and 4) g is finely continuous except perhaps on a semi-polar set. If we suppose also that the second semi-group is special standard and that the coexcessive function g is regular, then 1) g is a.s. right continuous and has left limits on the sample paths of the process \(\left( {X_t } \right)_{0 < t < \zeta }\), and 2) g is finely continuous except perhaps on a polar set.
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Weil, M. Propriétés de continuité fine des fonctions coexcessives. Z. Wahrscheinlichkeitstheorie verw Gebiete 12, 75–86 (1969). https://doi.org/10.1007/BF00538525
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DOI: https://doi.org/10.1007/BF00538525