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A zero-one law for integral functionals of the Bessel process. (English) Zbl 0704.60082
Séminaire de probabilités XXIV 1988/89, Lect. Notes Math. 1426, 137-153 (1990).
[For the entire collection see Zbl 0695.00024.]
Necessary and sufficient conditions are given for the finiteness of the integral functionals \(\int^{t}_{0}f(R_ s)ds\), \(0\leq t<\infty\), where \((R_ t\), \(t\geq 0)\) is an n-dimensional Bessel process defined as \(R_ t=| W_ t|\), with \((W_ t\), \(t\geq 0)\) a Brownian motion in \(R^ n\) starting at x. Two cases are considered: (i) \(n\geq 2\) and \(x=0\), and (ii) \(n=2\) and \(x\neq 0.\)
The conditions given are in the form of a zero-one law. They can be seen as a counterpart of results for the case \(n=1\) due to H. J. Engelbert and W. Schmidt [Stochastic differential systems, Proc. 3rd IFIP-WG7/1 Work. Conf., Visegrad/Hung. 1980, Lect. Notes Control. Inf. Sci. 36, 47-55 (1981; Zbl 0468.60077)]. A counter-example shows that the zero-one law fails when \(n>2\) and \(x\neq 0\).
Reviewer: M.Iosifescu

60J65 Brownian motion
60F20 Zero-one laws
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