Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0789.60061
Burdzy, Krzysztof
Excursion laws and exceptional points on Brownian paths. (English)
[A] Azéma, J. (ed.) et al., Séminaire de probabilités XXVII. Berlin: Springer-Verlag. Lect. Notes Math. 1557, 177-181 (1993). ISBN 3-540-57282-1/pbk

This note provides an example of exceptional points on the Brownian path which cannot be constructed by using a method of {\it S. Watanabe} [Proc. Int. Congr. Math., Warszawa 1983, Vol. 2, 1117-1124 (1984; Zbl 0565.60044)]. Specifically, let $X$ be a standard linear BM and $f:[0,\infty) \to [0,\infty)$ a continuous function. A fragment of path $\{X(s), s \in (t,t+\varepsilon)\}$ is called an excursion within $f$- boundaries if $\varepsilon>0$ and $\vert X(t+u)-X(t) \vert<\varepsilon$ for all $u \in (0,\varepsilon)$. If 0 is not the starting point of an excursion within $f$-boundaries a.s., we say that starting points of excursions within $f$-boundaries are exceptional. The author exhibits a function $f$ such that a.s.\par (1) there exist a.s. excursions within $f$-boundaries and their starting points are exceptional,\par (2) the expected lifetime of the excursion under the Brownian excursion law within $f$-boundaries is infinite.\par The latter essentially implies that one cannot splice together paths generated by a Poisson point process with characteristic measure the Brownian excursion law within $f$-boundaries, i.e. Watanabe's approach fails to construct such exceptional points. The author raises the question of whether there exists an increasing function $f$ for which (1) and (2) hold.
[J.Bertoin (Paris)]

MSC 2000:: *60J65 Brownian motion

Keywords: exceptional points on the Brownian path; excursions; Brownian excursion law; Poisson point process; Watanabe's approach

Citations: Zbl 0565.60044

Cited in: Zbl 0843.60072

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]