A class of anticipative tangent processes on the Wiener space (Q2748297)

The authors define a vectorial derivation over the Wiener space; derivation along an anticipative tangent process. This process is written as \(d\xi^\alpha(\tau)= a^\alpha_\beta dx^\alpha(\tau)+ c^\alpha d\tau\), where \(a^\alpha_\beta= -a^\beta_\alpha\), the skew symmetry is necessary to avoid second-order terms in the derivation process. For \(F\in W_{2,p'}\), they get NEWLINE\[NEWLINED_\xi F= \int^1_0 \Biggl(\sum_\alpha a^\alpha_\beta D_{\tau,\alpha} F\Biggr) dx^\beta(\tau)+ \sum_\alpha \int^1_0 c_\alpha D_{\tau,\alpha} F d\tau,NEWLINE\]NEWLINE and also an integration by parts formula. They apply this anticipative calculus to derive an integration by parts formula on the path space of a Riemannian manifold \(M\). The link is given by Theorem 2 which states that a smooth functional \(F\) defined on the path space \(P_{m_0}(M)\) is differentiable along a tangent vector field \(Z\) if and only if \(f\circ I\) is differentiable on the Wiener space along the tangent process \(d\xi= \dot z d\tau+ \rho dx(\tau)\) where \(d\rho(\tau)= \Omega(\circ dx,z)\), \(\Omega\) being the curvature tensor of \(M\).