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April 2001 Absolute Continuity of Heat Kernel Measure with Pinned Wiener Measure on Loop Groups
Bruce K. Driver, Vikram K. Srimurthy
Ann. Probab. 29(2): 691-723 (April 2001). DOI: 10.1214/aop/1008956690

Abstract

Let $t > 0, K$ be a connected compact Lie group equipped with an $Ad_K$- invariant inner product on the Lie Algebra of $K$. Associated to this data are two measures $\mu^0_t$ and $\nu^0_t$ on $\mathscr{L}(K)$ – the space of continuous loops based at $e \in K$. The measure $\mu^0_t$ is pinned Wiener measure with “variance $t$” while the measure $\nu^0_t$ is a “heat kernel measure” on $\mathscr{L}(K)$. The measure $\mu^0_t$ is constructed using a $K$-valued Brownian motion while the measure $\nu^0_t$ is constructed using a $\mathscr{L}(K)$-valued Brownian motion. In this paper we show that $\nu^0_t$ is absolutely continuous with respect to $\mu^0_t$ and the Radon-Nikodym derivative $d\nu^0_t /d\mu^0_t$ is bounded.

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Bruce K. Driver. Vikram K. Srimurthy. "Absolute Continuity of Heat Kernel Measure with Pinned Wiener Measure on Loop Groups." Ann. Probab. 29 (2) 691 - 723, April 2001. https://doi.org/10.1214/aop/1008956690

Information

Published: April 2001
First available in Project Euclid: 21 December 2001

zbMATH: 1018.60059
MathSciNet: MR1849175
Digital Object Identifier: 10.1214/aop/1008956690

Subjects:
Primary: 58D30 , 60H07
Secondary: 58D20

Keywords: Absolute continuity , heat kernel measures , Loop groups

Rights: Copyright © 2001 Institute of Mathematical Statistics

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Vol.29 • No. 2 • April 2001
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