CAMERON-MARTIN THEOREM

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

Contents

Motivation

Statement of the theorem

Integration by parts

Motivation

Recall that standard Gaussian measure $gamma^\{n\}$ on $mathbb\{R\}^\{n\}$ is not translation-invariant, but does satisfy the relation
::
★ } (gamma^{n})}{mathrm{d} gamma^{n}} (x) = exp left( langle h, x
angle_{mathbb{R}^{n}} - rac{1}{2} | h |_{mathbb{R}^{n}}^{2}
ight),
where the derivative on the left-hand side is the Radon-Nikodym derivative, and $(T\_\{h\})\_\{$
★ } (gamma^{n}) is the push forward of standard Gaussian measure by the translation map $T\_\{h\}\; :\; x\; mapsto\; x\; +\; h$.
Abstract Wiener measure $gamma$ on a separable Banach space $E$, where $i\; :\; H\; o\; E$ is an abstract Wiener space, is also "Gaussian" in some 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 $i(H)\; subseteq\; E$.

Statement of the theorem

Let $i\; :\; H\; o\; E$ be an abstract Wiener space with abstract Wiener measure $gamma\; :\; mathrm\{Borel\}\; (E)\; o\; [0,\; 1]$. For $h\; in\; H$, define $T\_\{h\}\; :\; E\; o\; E$ by $T\_\{h\}\; :\; x\; mapsto\; x\; +\; i(h)$. Then $(T\_\{h\})\_\{$
★ } (gamma) is equivalent to $gamma$ with Radon-Nikodym derivative
::
★ } (gamma)}{mathrm{d} gamma} (x) = exp left( langle h, x
angle^{sim} - rac{1}{2} | h |_{H}^{2}
ight),
where $langle\; h,\; x$
angle^{sim} = I(h) (x) denotes the Paley-Wiener integral.
It is important to note that the Cameron-Martin formula is only valid for translations by elements of the dense subspace $i(H)\; subseteq\; E$, and not by arbitrary elements of $E$. If the Cameron-Martin formula did hold for arbitrary translations, it would contradict the following result:
If $E$ is a separable Banach space and $mu$ is a locally finite Borel measure on $E$ that is equivalent to its own push forward under any translation, then either $E$ has finite dimension or $mu$ is the trivial (zero) measure. (See quasi-invariant measure.)
In fact, $gamma$ is quasi-invariant under $x\; mapsto\; x\; +\; v$ if and only if $v\; in\; i(H)$. Vectors in $i(H)$ are sometimes known as 'Cameron-Martin directions'.

Integration by parts

The Cameron-Martin formula gives rise to an integration by parts formula on $E$: if $F\; :\; E\; o\; mathbb\{R\}$ has bounded Fréchet derivative $mathrm\{D\}\; F\; :\; E\; o\; mathrm\{Lin\}\; (E;\; mathbb\{R\})\; =\; E^\{$
★ }, integrating the Cameron-Martin formula with respect to Wiener measure on both sides gives
::$int\_\{E\}\; F(x\; +\; t\; i(h))\; ,\; mathrm\{d\}\; gamma\; (x)\; =\; int\_\{E\}\; F(x)\; exp\; left(\; t\; langle\; h,\; x$
angle^{sim} - rac{1}{2} t^{2} | h |_{H}^{2}
ight) , mathrm{d} gamma (x) for any$t\; in\; mathbb\{R\}.$
Formally differentiating with respect to $t$ and evaluating at $t\; =\; 0$ gives the integration by parts formula
::$int\_\{E\}\; mathrm\{D\}\; F(x)\; (i(h))\; ,\; mathrm\{d\}\; gamma\; (x)\; =\; int\_\{E\}\; F(x)\; langle\; h,\; x$
angle^{sim} , mathrm{d} gamma (x)
Comparison with the divergence theorem of vector calculus suggests
::$mathrm\{div\}\; [V\_\{h\}]\; (x)\; =\; -\; langle\; h,\; x$
angle^{sim},
where $V\_\{h\}\; :\; E\; o\; E$ is the constant "vector field" $x\; mapsto\; i(h)$ for $x\; in\; E$. The wish to consider more general vector fields 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.