Volterra transformations of the Wiener measure on the space of continuous functions of two variables

Abstract

Tx(s, t) = x(s, t) + fTiΓfa, v)x(u, v)dudv , where the kernel K(u, v) is continuous. A stochastic integral analogous to K. Ito's is defined and used to determine a Jacobian J(x) for T such that if F{x) is a Wiener measurable functional, Γ a Wiener measurable set, and m Wiener measure, f F(x)dm = [ F(Tx)J(x)dm. JΓ JT~1(Γ)