In mathematics, the Cameron–Martin theorem or Cameron–Martin formula (named after Robert Horton Cameron and W. T. Martin) is a theorem of measure theory that describes how abstract Wiener measure changes under translation by certain elements of the Cameron–Martin Hilbert space.
(In fact, there is a unique translation invariant Radon measure up to scale by Haar's theorem: the
refers to the standard Euclidean dot product in
The Gaussian measure of the translation of
, the Gaussian measure scales by the distribution function appearing in the last display: The measure that associates to the set
is the pushforward measure, denoted
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
is an abstract Wiener space, is also a "Gaussian measure" in a suitable sense.
It turns out that a similar formula to the one above holds if we consider only translations by elements of the dense subspace
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
Consider a locally convex vector space
For an element in the topological dual
extended to the closure as Define the norm then the Cameron-Martin space
has bounded Fréchet derivative
, integrating the Cameron–Martin formula with respect to Wiener measure on both sides gives for any
gives the integration by parts formula Comparison with the divergence theorem of vector calculus suggests where
is the constant "vector field"
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.
Using Cameron–Martin theorem one may establish (See Liptser and Shiryayev 1977, p. 280) that for a
symmetric non-negative definite matrix
are continuous and satisfy the condition it holds for 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
, and we have the original formula as established by Cameron and Martin: