The Annals of Probability

Absolute Continuity of Heat Kernel Measure with Pinned Wiener Measure on Loop Groups

Bruce K. Driver and Vikram K. Srimurthy

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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 $\mathcal{L}(K)$ – 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 $\mathcal{L}(K)$. The measure $\mu^0_t$ is constructed using a $K$-valued Brownian motion while the measure $\nu^0_t$ is constructed using a $\mathcal{L}(K)$-valued Brownian motion. In this paper we show that $\nu^0_t$ is absolutely continuous with respect to $\mu^0_t$ and the Radon­Nikodym derivative $d\nu^0_t /d\mu^0_t$ is bounded.

Article information

Source
Ann. Probab., Volume 29, Number 2 (2001), 691-723.

Dates
First available in Project Euclid: 21 December 2001

Permanent link to this document
https://projecteuclid.org/euclid.aop/1008956690

Digital Object Identifier
doi:10.1214/aop/1008956690

Mathematical Reviews number (MathSciNet)
MR1849175

Zentralblatt MATH identifier
1018.60059

Subjects
Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus 58D30: Applications (in quantum mechanics (Feynman path integrals), relativity, fluid dynamics, etc.)
Secondary: 58D20: Measures (Gaussian, cylindrical, etc.) on manifolds of maps [See also 28Cxx, 46T12]

Keywords
Loop groups heat kernel measures absolute continuity

Citation

Driver, Bruce K.; Srimurthy, Vikram K. Absolute Continuity of Heat Kernel Measure with Pinned Wiener Measure on Loop Groups. Ann. Probab. 29 (2001), no. 2, 691--723. doi:10.1214/aop/1008956690. https://projecteuclid.org/euclid.aop/1008956690


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