A Weitzenbock formula for the damped Ornstein-Uhlenbeck operator in adapted differential geometry

On the Riemannian path space we consider the Ornstein-Uhlenbeck operator as sociated to the Dirichlet form epsilon (S, g) = E <<(<del>)over tilde> f, < (<del>)over tilde> g > (H), where <(<del>)over tilde> is the damped gradien t and <., .> (H) the scalar product of the Cameron-Martin space H. We prove a corresponding WeitzenbGck formula restricted to adapted vector fileds: t he Ricci-tensor is shown to be equal to the identity. (C) 2001 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.