Weak Levi-Civita Connection for the Damped Metric on the Riemannian Path Space and Vanishing of Ricci Tensor in Adapted Differential Geometry

2001

Abstract We shall establish in the context of adapted differential geometry on the path space P m o ( M ) a Weitzenbock formula which generalizes that in (A. B. Cruzeiro and P. Malliavin, J. Funct. Anal . 177 (2000), 219–253), without hypothesis on the Ricci tensor. The renormalized Ricci tensor will be vanished. The connection introduced in (A. B. Cruzeiro and S. Fang, 1997, J. Funct. Anal. 143 , 400–414) will play a central role.

A Weitzenböck formula for the damped Ornstein–Uhlenbeck operator in adapted differential geometry

2001

Abstract On the Riemannian path space we consider the Ornstein–Uhlenbeck operator associated to the Dirichlet form E (f,g)=E〈 ∇ f, ∇ g〉 H , where ∇ is the damped gradient and 〈·,·〉 H the scalar product of the Cameron–Martin space H . We prove a corresponding Weitzenbock formula restricted to adapted vector fileds: the Ricci-tensor is shown to be equal to the identity.

A probabilistic Weitzenböck formula on Riemannian path space

2000