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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

##### MSC:
 60J65 Brownian motion 60F20 Zero-one laws
##### Keywords:
Bessel process; Brownian motion; zero-one law; counter-example
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