The Construction of Filtrations on Abstract Wiener Space

The abstract Wiener space, unlike the classical Wiener space, does not possess a natural notion of time. The introduction of such a notion through a resolution of the identity on the Cameron–Martin space associated with the abstract Wiener space is considered in this paper. The notion of vector fields adapted to this filtration and causal linear operators are introduced and geometric characterizations of these notions are derived. The notion of equivalence between two abstract Wiener spaces endowed with filtrations is defined. It is shown that under some general conditions, every abstract Wiener space endowed with a filtration is equivalent to a classical Wiener space of a finite or countably infinite collection of independent one-dimensional Wiener processes. Letube an adapted vector field andδuthe divergence ofu; some sufficient conditions for the exponential integrability ofδuare derived.