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

✍ Scribed by Bruce K Driver

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 744 KB
• Volume: 149
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1997.3103

✦ Synopsis

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 time t, 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. 1997 Academic Press Contents 1. Introduction. 1.1 Statement of results. 2. Notation and prerequisites. 3. Brownian motion on loop groups. 3.1 L(g)-valued Bronian motion. 3.2. L(G)valued Brownian motion. 3.3. Generator of the process 7. 4. Integration by parts on the path space of L(G). 4.1. Parallel translation.

4.2. Integration by parts. 4.3. Closability of the Dirichlet form. 5. The finite dimensional approximations. 5.1. Finite dimensional integration by parts formula. 5.2. Geometry of the finite dimensional approximations. 6. Integration by parts on the loop group. 6.1. Passing to the limit. 7. Quasi-invariance of the heat kernel measure. 7.1. Finite dimensional preliminearies. 7.2. Quasi-invariance for the heat kernel measure on L(G). 8. Appendix: Review of the Ito^integral in infinite dimensions. 8.1. Continuous Hilbert valued local martingales. 8.2. The Ito^integral on our abstract Wiener space. 8.3. Backwards Ito^integrals.