| Let K be a simply-connected compact Lie Group equipped with an AdK-invariant inner product on the Lie Algebra K , of K. Given this data, there is a well known left invariant "H1-Riemannian structure" on L (K) (the infinite dimensional group of continuous based loops in K), as well as a heat kernel nT (k0,·) associated with the Laplace-Beltrami operator on L (K). Here T > 0, k0 ∈ L (K), and nT (k0,·) is a certain probability measure on L (K). In this paper we show that Heat Kernel measure n1 (e,·) is absolutely continuous with respect to Pinned Wiener Measure on K and that the two measures are equivalent on Gs0≡ s ⟨xt : t ∈ [0, s0]⟩ (the s -algebra generated by truncated loops up to "time" s0). |
