A change of scale formula for conditional Wiener integrals on classical Wiener space

Let X_k(x) = (∫₀^T α ₁(s)dx(s),..., ∫₀^T α_k(s) dx(s)) and X_τ(x) = (x(t₁), ..., x(t_k)) on the classical Wiener space, where {α₁,...,α_k} is an orthonormal subset of L₂[0,T] and τ : 0 < t₁ < ⋯ < t_k = T is a partition of [0,T]. In this paper, we establish a change of scale formula for conditional Wiener integrals E[G _τ|X_k] of functions on classical Wiener space having the form G_r(x) = F(x)Ψ(∫₀^T v ₁(s)dx(s),..., ∫₀^T v_r(s)dx(s)), for F ∈ S and Ψ = ψ + Φ (ψ ∈ L_p(ℝ ^r), Φ ∈M̂(ℝ^r)), which need not be bounded or continuous. Here S is a Banach algebra on classical Wiener space and M̂(ℝ^r) is the space of Fourier transforms of measures of bounded variation over ℝ^r. As results of the formula, we derive a change of scale formula for the conditional Wiener integrals E[G _r|X_τ] and B[F|X_τ]. Finally, we show that the analytic Feynman integral of F can be expressed as a limit of a change of scale transformation of the conditional Wiener integral of F using an inversion formula which changes the conditional Wiener integral of F to an ordinary Wiener integral of F, and then we obtain another type of change of scale formula for Wiener integrals of F.