The concept of an abstract Wiener space is a mathematical construction developed by
Leonard Gross
Leonard Gross (born February 24, 1931) is an American mathematician and Professor Emeritus of Mathematics at Cornell University.
Gross has made fundamental contributions to mathematics and the mathematically rigorous study of quantum field theo ...
to understand the structure of
Gaussian measure
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space R''n'', closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are nam ...
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 i ...
is the prototypical example.
The
structure theorem for Gaussian measures In mathematics, the structure theorem for Gaussian measures shows that the abstract Wiener space construction is essentially the only way to obtain a strictly positive Gaussian measure on a separable Banach space. It was proved in the 1970s by Kal ...
states that all Gaussian measures can be represented by the abstract Wiener space construction.
Motivation
Let
be a real
Hilbert space, 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
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Mea ...
on the original Hilbert space
. On the other hand, suppose
is a Banach space that contains
as a dense 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