Analogues of conditional Wiener integrals and their change of scale transformations on a function space

For a partition 0 = t 0 < t 1 < ⋯ < t n < t n + 1 = t of the interval [ 0 , t ] , let X n ( x ) = ( x ( t 0 ) , … , x ( t n ) ) and X n + 1 ( x ) = ( x ( t 0 ) , … , x ( t n ) , x ( t n + 1 ) ) on a generalized Wiener space ( C [ 0 , t ] , B ( C [ 0 , t ] ) , w φ ) , where C [ 0 , t ] is the space of the continuous paths on [ 0 , t ] and w φ is a probability measure on the Borel class B ( C [ 0 , t ] ) of C [ 0 , t ] . In this paper, using simple formulas for conditional w φ -integrals on C [ 0 , t ] with the conditioning functions X n and X n + 1 , we evaluate the conditional analytic Wiener w φ -integrals of the function G r having the form G r ( x ) = F ( x ) Ψ ( ∫ 0 t v 1 ( s ) d x ( s ) , … , ∫ 0 t v r ( s ) d x ( s ) ) , for F ∈ S w φ which is a Banach algebra, and for Ψ = f + ϕ which need not be bounded or continuous, where f ∈ L p ( R r ) and ϕ ∈ M ˆ ( R r ) . And also, we establish several change of scale transformations for the conditional analytic Wiener w φ -integrals of G r with the conditioning functions X n and X n + 1 .