Logarithmic Sobolev inequalities for diffusion Processes with application to path space

Let （M, g） be a connected Riemannian manifold and let L = 1/2（Δ+ Z） for some C¹-vector field Z. This paper uses Kendall’s coupling analysis to obtain an estimation of the logarithmic Sobolev （abbrev. L.S.） constant with respect to the distribution of the L-diffusion process at time t, which then is used to prove a L. S. inequality on the path space. The main result can be considered as an extension of 1 in which Z is taken to be zero. Moreover, as a generalization to 2; Theorem 1.5 and 3; Theorem 1 which were proved for diffusions on R^d, an upper bound estimation of L.S. constant for the L-diffusion process is also presented.