Universal Connection , Stochastic Differential Equation , Calculus on Loop Space

where FCb (Lx(M)) denotes the set of the smooth cylindrical functions and ∇(i) denotes the covariant derivative with respect to the i−th variable. Also τ(γ)t : TxM → Tγ(t)M is the stochastic parallel translation along γ. Among Riemannian connections, we restrict ourselves to the torsion skew symmetric (=TSS in short) connection, Namely the torsion T must satisfy that g(T (X, Y ), Y ) = 0. for any vector fields X, Y . Then for the associated derivative D on Lx(M) is a closable operator on L(Lx(M), νx). This is an immediate consequence of the integration by parts formula [D]. We denote the closure by the same notation D and the domain of the smallest closed extension by D. Next we consider the topology on Lx(M). Let us assume that M is isometrically embedded into the Euclidean space R. As is well-known, νx(W∩Lx(M)) = 1. Here