Logarithmic Sobolev inequality on free path space over a compact Riemannian manifold

In this paper we use Bismut’s method to verify the formula of integration by parts on the free path space. By the formula, we verify the martingale representation for reference measure PμPμ, the law of the Brownian motion on the base manifold with initial distribution μμ. Then by the martingale representation we get the logarithmic Sobolev inequality on compact free Riemannian path space.