The concept of an abstract Wiener space is a mathematical construction developed by
Leonard Gross to understand the structure of
Gaussian measure
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space \mathbb^n, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are na ...
s on infinite-dimensional spaces. The construction emphasizes the fundamental role played by the
Cameron–Martin space. The
classical Wiener space
In mathematics, classical Wiener space is the collection of all continuous functions on a given domain (usually a subinterval of the real line), taking values in a metric space (usually ''n''-dimensional Euclidean space). Classical Wiener space ...
is the prototypical example.
The
structure theorem for Gaussian measures states that all Gaussian measures can be represented by the abstract Wiener space construction.
Motivation
Let
be a real
Hilbert space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
, assumed to be infinite dimensional and
separable. In the physics literature, one frequently encounters integrals of the form
:
where
is supposed to be a normalization constant and where
is supposed to be the
non-existent Lebesgue measure on
. Such integrals arise, notably, in the context of the
Euclidean path-integral formulation of quantum field theory. At a mathematical level, such an integral cannot be interpreted as integration against a
measure on the original Hilbert space
. On the other hand, suppose
is a Banach space that contains
as a
dense
Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be use ...
subspace. If
is "sufficiently larger" than
, then the above integral can be interpreted as integration against a well-defined (Gaussian) measure on
. In that case, the pair
is referred to as an abstract Wiener space.
The prototypical example is the classical Wiener space, in which
is the Hilbert space of real-valued functions
on an interval