"Absolute Continuity of Heat Kernel Measure with Pinned Wiener Measure on Loop groups" (Joint with Vikram
Srimurthy)
.
(UCSD Preprint Apri 2000, To appear in Annals of probability. )
Available as a DVI
file or a PDF file.
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 $\mu _{t}^{0}$ and $\nu _{t}^{0}$ on $\mathcal{L}(K)$ -- the space of continuous loops based at $e\in K.$ The measure $\mu _{t}^{0}$ is pinned Wiener measure with ``variance $t$'' while the measure $\nu _{t}^{0}$ is a ``heat kernel measure'' on $\mathcal{L}(K).$ The measure $\mu _{t}^{0}$ is constructed using a $K$ -- valued Brownian motion while the measure $\nu _{t}^{0}$ is constructed using a $\mathcal{L}(K)$ -- valued Brownian motion. In this paper we show that $\nu _{t}^{0}$ is absolutely continuous with respect to $\mu _{t}^{0}$ and the Radon-Nikodym derivative $% d\nu _{t}^{0}/d\mu _{t}^{0}$ is bounded.
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