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

Shigeki Aida

Research output: Contribution to journalArticlepeer-review

14 Citations (Scopus)

Abstract

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 Hn 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 Hn, we will prove an LSI on each homotopy class of the loop space over a constant negative curvature compact Riemannian manifold.

Original languageEnglish
Pages (from-to)430-477
Number of pages48
JournalJournal of Functional Analysis
Volume174
Issue number2
DOIs
Publication statusPublished - 2000 Jul 10

ASJC Scopus subject areas

  • Analysis

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