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The gauge and conditional gauge theorem. (English) Zbl 0561.60084

Sémin de probabilités XIX, Univ. Strasbourg 1983/84, Proc., Lect. Notes Math. 1123, 496-503 (1985).
[For the entire collection see Zbl 0549.00007.]
This is mainly a note on the gauge and conditional gauge theorem [cf. the author and K. M. Rao, Stochastic processes, Semin. Evanston/Ill. 1981, Progr. Probab. Stat. 1, 1-29 (1981; Zbl 0492.60073) and N. Falkner, Z. Wahrscheinlichkeitstheorie. Verw. Geb. 65, 19-33 (1983; Zbl 0496.60078)]. The author proves that the gauge theorem holds true if we extend the class of bounded functions to the Kato class \[ K_ d=\{g:\lim_{t\downarrow 0}\sup_{x\in D}E^ x[\int^{t}_{0}1_ D| g| (X_ s)ds=0,\quad D\subset {\mathbb{R}}^ d\} \] and that the conditional gauge theorem follows rather quickly from the gauge theorem if the boundary of the domain is Lipschitzian.
Reviewer: Ch.Wu

MSC:

60J65 Brownian motion
60H05 Stochastic integrals
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