Consider a Stratonovich stochastic differential equation

dχt=X(χt)odBt+A(χt)dt
(1.1)
with C∞ coefficients on a compact Riemannian manifold M, with associated differential generator A=12ΔM+Z and solution flow {ξt : t ≥ 0} of random smooth diffeomorphisms of M. Let Tξt: TM → TM be the induced map on the tangent bundle of M obtained by differentiating ξt with respect to the initial point. Following an observation by A. Thalmaier we extend the basic formula of [EL94] to obtain

Edf(Tξ.(h.))=EF(ξ.(χ))∫T0⟨Tξs(h˙s),X(ξs(χ))dBs⟩
(1.2)
where F∈FC∞b(Cχ(M)), the space of smooth cylindrical functions on the space C x (M) of continuous paths γ : [0,T] → M with γ(0) = x, dF is its derivative, and h. is a suitable adapted process with sample paths in the Cameron-Martin space L
2,1
0
([0,T];T x M).Set F
x
t
= σ{ξs(x) : 0 ≤ s ≤ t} Taking conditional expectation with respect to.F
x
T
, formula (1.2) yields integration by parts formulae on C x (M) of the form

EdF(γ)(V¯¯¯¯h)=EF(γ)δV¯¯¯¯h(γ)
(1.3)
where V¯¯¯¯h is the vector field on C x(M)

V¯¯¯¯h(γ)t−E{Tξt(ht)|ξ.(χ)=γ}
and δV¯¯¯¯h:Cx(M)→ is given by

δV¯¯¯¯h(γ)=IE{∫T0<Tξs(h˙s),X(ξs(x))dBs>|ξ.(x)=γ}