"On the Equivalence of Measures on Loop Space" by Vikram Srimurthy.(UCSD Thesis, June 1999) Abstract Let $K$ be a simply-connected compact Lie Group equipped with an $Ad_{K}$% -invariant inner product on the Lie Algebra $\frak{K}$, of $K$. Given this data, there is a well known left invariant ''$H^{1}$-Riemannian structure'' on $L\left( K\right) $ (the infinite dimensional group of continuous based loops in K), as well as a heat kernel $\nu _{T}\left( k_{0},\cdot \right) $associated with the Laplace-Beltrami operator on $L\left( K\right) $. Here $% T>0$, $k_{0}\in L\left( K\right) $, and $\nu _{T}\left( k_{0},\cdot \right) $ is a certain probability measure on $L\left( K\right) $. In this paper we show that $\nu _{1}\left( e,\cdot \right) $ is equivalent to Pinned Wiener Measure on $K$ on $\frak{G}_{s_{0}}\equiv \sigma \left\langle x_{t}:t\in \left[ 0,s_{0}\right] \right\rangle $ (the $\sigma $-algebra generated by truncated loops up to ``time'' $s_{0}$) June, 1999 The following two papers constitute (together) a refined version of Srimurthy's is thesis.
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08/15/2016 02:53 PM |