TY - JOUR

T1 - Logarithmic Derivatives of Heat Kernels and Logarithmic Sobolev Inequalities with Unbounded Diffusion Coefficients on Loop Spaces

AU - Aida, Shigeki

N1 - Funding Information:
1This research was partially supported by The Inamori Foundation.

PY - 2000/7/10

Y1 - 2000/7/10

N2 - 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.

AB - 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.

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U2 - 10.1006/jfan.2000.3592

DO - 10.1006/jfan.2000.3592

M3 - Article

AN - SCOPUS:0000448404

SN - 0022-1236

VL - 174

SP - 430

EP - 477

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 2

ER -