Integration by Parts and Quasi-Invariance for Heat Kernel Measures on Loop Groups

Integration by parts formulas are established both for Wiener measure on the path space of a loop group and for the heat kernel measures on the loop group. The Wiener measure is defined to be the law of a certain loop group valued “Brownian motion” and the heat kernel measures are timet,t>0, distributions of this Brownian motion. A corollary of either of these integrations by parts formulas is the closability of the pre-Dirichlet form considered by B. K. Driver and T. Lohrenz [1996,J. Functional Anal.140, 381–448]. We also show that the heat kernel measures are quasi-invariant under right under right and left translations by finite energy loops.