## The Annals of Probability

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

#### 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

https://projecteuclid.org/euclid.aop/1008956690

Digital Object Identifier
doi:10.1214/aop/1008956690

Mathematical Reviews number (MathSciNet)
MR1849175

Zentralblatt MATH identifier
1018.60059

#### 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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