Cameron–Martin theorem

In mathematics, the Cameron–Martin theorem or Cameron–Martin formula is a theorem of measure theory that describes how Abstract [Wiener space|abstract Wiener measure] changes under translation by certain elements of the Cameron–Martin Hilbert space.

Motivation

The standard Gaussian measure on -dimensional Euclidean space is not translation-invariant. Instead, a measurable subset has Gaussian measure
Here refers to the standard Euclidean dot product in. The Gaussian measure of the translation of by a vector is
So under translation through, the Gaussian measure scales by the distribution function appearing in the last display:
The measure that associates to the set the number is the pushforward measure, denoted Here refers to the translation map:. The above calculation shows that the Radon–Nikodym derivative of the pushforward measure with respect to the original Gaussian measure is given by
The abstract Wiener measure on a separable Banach space, where is an abstract Wiener space, is also a "Gaussian measure" in a suitable sense. How does it change under translation? It turns out that a similar formula to the one above holds if we consider only translations by elements of the dense subspace.

Statement of the theorem

For abstract Wiener spaces

Let be an abstract Wiener space with abstract Wiener measure. For, define by. Then is equivalent to with Radon–Nikodym derivative
where
denotes the Paley–Wiener integral.
The Cameron–Martin formula is valid only for translations by elements of the dense subspace, called Cameron–Martin space, and not by arbitrary elements of. If the Cameron–Martin formula did hold for arbitrary translations, it would contradict the following result:
In fact, is quasi-invariant under translation by an element if and only if. Vectors in are sometimes known as Cameron–Martin directions.

Version for locally convex vector spaces

Consider a locally convex vector space, with a Gaussian measure on the cylindrical σ-algebra and let denote the translation by. For an element in the topological dual define the distance to the mean
and denote the closure in as.
Define the covariance operator extended to the closure as
Define the norm
then the Cameron–Martin space of in is
If for there exists an such that then and. Further there is equivalence with Radon-Nikodým density
If the two measures are singular.

Integration by parts

The Cameron–Martin formula gives rise to an integration by parts formula on : if has bounded Fréchet derivative, integrating the Cameron–Martin formula with respect to Wiener measure on both sides gives
for any. Formally differentiating with respect to and evaluating at gives the integration by parts formula
Comparison with the divergence theorem of vector calculus suggests
where is the constant "vector field" for all. The wish to consider more general vector fields and to think of stochastic integrals as "divergences" leads to the study of stochastic processes and the Malliavin calculus, and, in particular, the Clark–Ocone theorem and its associated integration by parts formula.

An application

Using Cameron–Martin theorem one may establish that for a symmetric non-negative definite matrix whose elements are continuous and satisfy the condition
it holds for a −dimensional Wiener process that
where is a nonpositive definite matrix which is a unique solution of the matrix-valued Riccati differential equation
with the boundary condition.
In the special case of a one-dimensional Brownian motion where, the unique solution is, and we have the original formula as established by Cameron and Martin: