Various topologies in the Wiener space and Lévy's stochastic area

Abstract

In this paper, we observe how Levy's stochastic area looks when we see it through various topologies in the Wiener space. Our theorem implies that it is quite natural from the viewpoint of topology to define a distinct skeleton of Levy's stochastic areaS(w) for each distinct topology in the Wiener space, or equivalently, for each distinct abstract Wiener space on which the Wiener measure andS(w) are realized. Thus we cannot determine its intrinsic skeleton in the theory of abstract Wiener spaces.