The theory of infinite dimensional oscillatory integrals and some of its applications are discussed, with special attention to the relations with the original work of K. Itô in this area. A recent general approach to infinite dimensional integration which unifies the case of oscillatory integrals and the case of probabilistic type integrals is presented, together with some new developments.
"Infinite dimensional oscillatory integrals as projective systems of functionals." J. Math. Soc. Japan 67 (4) 1295 - 1316, October, 2015. https://doi.org/10.2969/jmsj/06741295