In mathematics, the CameronâMartin theorem or CameronâMartin 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 CameronâMartin Hilbert space.
The standard Gaussian measure on -dimensional Euclidean space is not translation- invariant. (In fact, there is a unique translation invariant Radon measure up to scale by Haar's theorem: the -dimensional Lebesgue measure, denoted here .) 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 RadonâNikodym 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 .
Let be an abstract Wiener space with abstract Wiener measure . For , define by . Then is equivalent to with RadonâNikodym derivative
where
denotes the PaleyâWiener integral.
The CameronâMartin formula is valid only for translations by elements of the dense subspace , called CameronâMartin space, and not by arbitrary elements of . If the CameronâMartin 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 CameronâMartin directions.
The CameronâMartin formula gives rise to an integration by parts formula on : if has bounded FrĂ©chet derivative , integrating the CameronâMartin 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 ClarkâOcone theorem and its associated integration by parts formula.
Using CameronâMartin theorem one may establish (See Liptser and Shiryayev 1977, p. 280) that for a symmetric non-negative definite matrix whose elements are continuous and satisfy the condition
it holds for a âdimensional 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:
In mathematics, the CameronâMartin theorem or CameronâMartin 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 CameronâMartin Hilbert space.
The standard Gaussian measure on -dimensional Euclidean space is not translation- invariant. (In fact, there is a unique translation invariant Radon measure up to scale by Haar's theorem: the -dimensional Lebesgue measure, denoted here .) 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 RadonâNikodym 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 .
Let be an abstract Wiener space with abstract Wiener measure . For , define by . Then is equivalent to with RadonâNikodym derivative
where
denotes the PaleyâWiener integral.
The CameronâMartin formula is valid only for translations by elements of the dense subspace , called CameronâMartin space, and not by arbitrary elements of . If the CameronâMartin 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 CameronâMartin directions.
The CameronâMartin formula gives rise to an integration by parts formula on : if has bounded FrĂ©chet derivative , integrating the CameronâMartin 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 ClarkâOcone theorem and its associated integration by parts formula.
Using CameronâMartin theorem one may establish (See Liptser and Shiryayev 1977, p. 280) that for a symmetric non-negative definite matrix whose elements are continuous and satisfy the condition
it holds for a âdimensional 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: