Logarithmic Derivatives of Heat Kernels and Logarithmic Sobolev Inequalities with Unbounded Diffusion Coefficients on Loop Spaces

✍ Scribed by Shigeki Aida

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 294 KB
• Volume: 174
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2000.3592

✦ Synopsis

dedicated to professor norio shimakura on the occasion of his sixtieth birthday

In this paper, we will give a sufficient condition on the logarithmic derivative of the heat kernel under which a logarithmic Sobolev inequality (LSI, in abbreviation) on a loop space holds. As an application, we prove an LSI on a pinned path space over the hyperbolic space H n with constant sectional curvature &a (a 0). The diffusion coefficient of the Dirichlet form is an unbounded but exponentially integrable function. Applying to the case when a=0, we can prove an LSI with a logarithmic Sobolev constant 18 in the case of standard pinned Brownian motion. Using the LSI on the pinned path space on H n , we will prove an LSI on each homotopy class of the loop space over a constant negative curvature compact Riemannian manifold.

2000 Academic Press

| 0 F 2 log(F 2 Â&F& 2 ) dP(#) | 0 C(#) |DF(#)| 2 dP(#),

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