Integration by parts formula and logarithmic Sobolev inequality on the path space over loop groups

The geometric stochastic analysis on the Riomannian path space developed re cently gives rise to the concept of tangent processes. Roughly speaking, it is the infinitesimal version of the Girsanov theorem. Using this concept, we shall establish a formula of integration by parts on the path space over a loop group. Following the martingale method developed in Capitaine, Hsu and Ledoux, we shall prove that the logarithmic Sobolev inequality holds on the full paths. As a particular case of our result, vue obtain the Driver- Lohrenz's heat kernel logarithmic Sobolev inequalities over loop groups. Th e stochastic parallel transport introduced by Driver will play a crucial ro le.