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Brownian intersection local times: Exponential moments and law of large masses

Author(s): Wolfgang König; Peter Mörters
Journal: Trans. Amer. Math. Soc. 358 (2006), 1223-1255.
MSC (2000): Primary 60J65, 60J55, 60F10
Posted: May 9, 2005
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Abstract: Consider $p$ independent Brownian motions in $\mathbb{R} ^d$, each running up to its first exit time from an open domain $B$, and their intersection local time $\ell$ as a measure on $B$. We give a sharp criterion for the finiteness of exponential moments,

\begin{displaymath}\mathbb{E}\Big[\exp\Big(\sum_{i=1}^n \langle\varphi_i, \ell \rangle^{1/p}\Big) \Big],\end{displaymath}

where $\varphi_1, \dots, \varphi_n$ are nonnegative, bounded functions with compact support in $B$. We also derive a law of large numbers for intersection local time conditioned to have large total mass.

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Additional Information:

Wolfgang König
Affiliation: Institut für Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany
Address at time of publication: Mathematical Institute, University Leipzig, Augustusplatz 10/11, 04109 Leipzig, Germany
Email: koenig@math.tu-berlin.de, koenig@math.uni-leipzig.de

Peter Mörters
Affiliation: Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom
Email: maspm@bath.ac.uk

DOI: 10.1090/S0002-9947-05-03744-X
PII: S 0002-9947(05)03744-X
Keywords: Intersection of Brownian paths, intersection local time, exponential moment, Feynman-Kac formula
Received by editor(s): August 13, 2003
Received by editor(s) in revised form: May 4, 2004
Posted: May 9, 2005
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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