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Comparaison entre temps d’atteinte et temps de sejour de certaines diffusions réelles. (French) Zbl 0562.60085
Sémin. de probabilités XIX, Univ. Strasbourg 1983/84, Proc., Lect. Notes Math. 1123, 291-296 (1985).
[For the entire collection see Zbl 0549.00007.]
Let X and Y be Brownian motions with values in $${\mathbb{R}}^ d$$ and $${\mathbb{R}}^{d+2}$$ respectively, started from zero. Z. Ciesielski and S. Taylor [Trans. Am. Math. Soc. 103, 434-450 (1962; Zbl 0121.130)] noticed that the law of time spent by Y in the unit ball of $${\mathbb{R}}^{d+2}$$ is the same as the law of the first exit time from the unit ball of $${\mathbb{R}}^ d$$ by X.
This result is seen to hold for many couples of diffusions of the real line, the proof being computation of the Laplace transform of the laws involved. Some special cases are then explained by path transformations of the diffusions, using D. William’s decompositions [Proc. Lond. Math. Soc., III. Ser. 28, 738-768 (1974; Zbl 0326.60093)].

##### MSC:
 60J60 Diffusion processes
##### Keywords:
sojourn time; first exit time; path transformations
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