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

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.