Cameron-Martin theorem

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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.

[edit] Motivation

Recall that standard Gaussian measure $γ n$ on $\mathbb{R}^{n}$ is not translation-invariant, but does satisfy the relation

$\frac{\mathrm{d} (T_{h})_{*} (\gamma^{n})}{\mathrm{d} \gamma^{n}} (x) = \exp \left( \langle h, x \rangle_{\mathbb{R}^{n}} - \frac{1}{2} \| h \|_{\mathbb{R}^{n}}^{2} \right),$

where the derivative on the left-hand side is the Radon-Nikodym derivative, and $(T h) * (γ n)$ is the push forward of standard Gaussian measure by the translation map $T_{h} : x \mapsto x + h$ .

Abstract Wiener measure $γ$ on a separable Banach space $E$ , where $i : H \to E$ 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 $i(H) \subseteq E$ .

[edit] Statement of the theorem

Let $i : H \to E$ be an abstract Wiener space with abstract Wiener measure $\gamma : \mathrm{Borel} (E) \to [0, 1]$ . For $h \in H$ , define $T_{h} : E \to E$ by $T_{h} : x \mapsto x + i(h)$ . Then $(T h) * (γ)$ is equivalent to $γ$ with Radon-Nikodym derivative

$\frac{\mathrm{d} (T_{h})_{*} (\gamma)}{\mathrm{d} \gamma} (x) = \exp \left( \langle h, x \rangle^{\sim} - \frac{1}{2} \| h \|_{H}^{2} \right),$

where $\langle h, x \rangle^{\sim} = I(h) (x)$ 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 $μ$ 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 $μ$ is the trivial (zero) measure. (See quasi-invariant measure.)

In fact, $γ$ 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.

[edit] Integration by parts

The Cameron-Martin formula gives rise to an integration by parts formula on $E$ : if $F : E \to \mathbb{R}$ has bounded Fréchet derivative $\mathrm{D} F : E \to \mathrm{Lin} (E; \mathbb{R}) = E^{*}$ , integrating the Cameron-Martin formula with respect to Wiener measure on both sides gives

$\int_{E} F(x + t i(h)) \, \mathrm{d} \gamma (x) = \int_{E} F(x) \exp \left( t \langle h, x \rangle^{\sim} - \frac{1}{2} t^{2} \| h \|_{H}^{2} \right) \, \mathrm{d} \gamma (x)$ for any $t \in \mathbb{R}.$

Formally differentiating with respect to $t$ and evaluating at $t = 0$ gives the integration by parts formula

$\int_{E} \mathrm{D} F(x) (i(h)) \, \mathrm{d} \gamma (x) = \int_{E} F(x) \langle h, x \rangle^{\sim} \, \mathrm{d} \gamma (x)$

Comparison with the divergence theorem of vector calculus suggests

$\mathrm{div} [V_{h}] (x) = - \langle h, x \rangle^{\sim},$

where $V_{h} : E \to E$ 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.