The concept of an abstract Wiener space is a mathematical construction developed by Leonard Gross to understand the structure of Gaussian measures on infinite-dimensional spaces.
The construction emphasizes the fundamental role played by the Cameron–Martin space.
The structure theorem for Gaussian measures states that all Gaussian measures can be represented by the abstract Wiener space construction.
be a real Hilbert space, assumed to be infinite dimensional and separable.
In the physics literature, one frequently encounters integrals of the form where
Such integrals arise, notably, in the context of the Euclidean path-integral formulation of quantum field theory.
is the Wiener measure describing Brownian motion starting at the origin.
What the preceding example means is that we have a formal expression for the Wiener measure given by
Although this formal expression suggests that the Wiener measure should live on the space of paths for which
, this is not actually the case, as sample Brownian paths are known to be almost surely nowhere differentiable, though it can be generalized to random measures as tempered distributions through the characteristic function as the white noise measure.
Gross's abstract Wiener space construction abstracts the situation for the classical Wiener space and provides a necessary and sufficient (if sometimes difficult to check) condition for the Gaussian measure to exist on
As Gross himself puts it[1] (adapted to our notation), "However, it only became apparent with the work of I.E. Segal dealing with the normal distribution on a real Hilbert space, that the role of the Hilbert space
itself was auxiliary for many of Cameron and Martin's theorems, and in some instances even unnecessary."
One of the appealing features of Gross's abstract Wiener space construction is that it takes
appearing earlier in this section are purely formal, physics-style expressions, they are very useful in helping to understand properties of
be a Hilbert space defined over the real numbers, assumed to be infinite dimensional and separable.
is a set defined in terms of the values of a finite collection of linear functionals on
In that case, we can associate to the above-defined cylinder set
is called the standard Gaussian cylinder set measure on
One can understand the difficulty by considering the behavior of the standard Gaussian measure on
The expectation value of the squared norm with respect to this measure is computed as an elementary Gaussian integral as
That is, the typical distance from the origin of a vector chosen randomly according to the standard Gaussian measure on
(The typical distance from the origin would be infinite, so that the measure would not actually live on the space
is an injective continuous linear map whose image is dense in
The upshot of this whole discussion is that Gaussian integrals of the sort described in the motivation section do have a rigorous mathematical interpretation, but they do not live on the space whose norm occurs in the exponent of the formal expression.
The abstract Wiener space construction is not simply one method of building Gaussian measures.
Rather, every Gaussian measure on an infinite-dimensional Banach space occurs in this way.
Absolutely continuous paths starting at 0 with square-integrable first derivative
{\displaystyle H:=L_{0}^{2,1}([0,T];\mathbb {R} ^{n}):=\{{\text{Absolutely continuous paths starting at 0 with square-integrable first derivative}}\}}
in this case reflects the roughness of the typical Brownian path, which is known to be nowhere differentiable.