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

Authors: Wolfgang König and Peter Mörters
Journal: Trans. Amer. Math. Soc. 358 (2006), 1223-1255
MSC (2000): Primary 60J65, 60J55, 60F10
Published electronically: May 9, 2005
MathSciNet review: 2187652
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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

Peter Mörters
Affiliation: Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom

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
Published electronically: May 9, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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