Transformation of Measure Under Anticipative Flows

Abstract

In this chapter we shall study the absolute continuity of the image of Wiener measure under the flows defined by an ordinary differential equation defined on the Wiener space. The main point is that the vector fields defining these flows will be the elements of the Cameron martin space valued Sobolev spaces (i.e., D _p , _k (H)), hence they are equivalence classes with respect to the Wiener measure µ of random variables. Consequently to give a sense to the equations by which they are defined we need to know that the eventual flow solutions map the Wiener measure to an absolutely continuous measure (with respect to µ). One way to circumvent this difficulty is to obtain them as the limits of finite dimensional flows. For this we shall need some dimension independent majorations about the L ^p-moments of the Radon-Nikodym densities in the finite dimensional case. This will be done in this section. In the next two sections we shall pass to the infinite dimensional case and the fourth section will be devoted to a singular case in the particular situation of W = C ₀ ([0, 1], ℝ^d), where the vector field will not be H-valued but its Lebesgue integral will be with values in H.