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

Driver, Bruce K. ; Srimurthy, Vikram K.

Ann. Probab., Tome 29 (2001) no. 1, p. 691-723 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 2001-04-14
Classification:  Loop groups,  heat kernel measures,  absolute continuity,  60H07,  58D30,  58D20