Measure and integration Measures on topological spaces Theorem

Cameron–Martin theorem

Builds on Classical Wiener space Wiener process Gaussian measure Pushforward measure Absolutely continuous measure Mutually singular measures Radon–Nikodym derivative

Statement

Let $T > 0$ and let $C_0 = C_0([0, T])$ be the classical Wiener space of continuous functions $\omega \colon [0, T] \to \R$ with $\omega(0) = 0$ . Let $W$ be Wiener measure on $C_0$ , the law of a Wiener process on $[0, T]$ . Let $H$ be the Cameron–Martin space: the set of absolutely continuous $h \in C_0$ with $h' \in L^2([0, T])$ . For $h \in C_0$ , let $W_h$ be the pushforward of $W$ under the translation $\omega \mapsto \omega + h$ .

If $h \in H$ , then $W_h$ and $W$ are mutually absolutely continuous, and for $W$ -almost every $\omega$ $\frac{dW_h}{dW}(\omega) = \exp\left( \int_0^T h'(t) \, d\omega(t) - \frac12 \int_0^T h'(t)^2 \, dt \right).$
If $h \notin H$ , then $W_h$ and $W$ are mutually singular.

Remarks

Here $dW_h / dW$ is the Radon–Nikodym derivative, and $\int_0^T h'(t) \, d\omega(t)$ is the Wiener integral of the deterministic function $h'$ , defined as an $L^2(W)$ limit. In one dimension the formula is a direct computation: the density of $N(a, 1)$ with respect to $N(0, 1)$ is $e^{a x - a^2/2}$ . The general case reduces to finite-dimensional projections of the path, since the Wiener integral is a normal variable with variance $\int_0^T h'(t)^2 \, dt$ .

Example: for $h(t) = c t$ , one has $h' = c$ , and the density is $\exp(c\, \omega(T) - c^2 T / 2)$ . So Brownian motion with drift $c$ has a law equivalent to that of Brownian motion on $[0, T]$ .

The analogous result holds for every centered Gaussian measure on a separable Banach space, with $H$ its Cameron–Martin space. In infinite dimensions $H$ has measure zero, so most translations give singular measures.

Going deeper

For a student

The theorem is the starting point of Girsanov's theorem, where the deterministic shift is replaced by an adapted random drift. It shows that Wiener measure is quasi-invariant only along the directions of $H$ , which is a dense subspace of measure zero. Integration by parts on Wiener space, the base of Malliavin calculus, comes from differentiating the density in $h$ .

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Quasi-invariant measure

Tag	`WAE4ZGJXJ7`
URI	`m2://WAE4ZGJXJ7`
Kind	statement, theorem
Writing	W0 stub
MSC	28C20, 60G15, 60J65
Status	This copy of the site has no build data. Run `m2 status WAE4ZGJXJ7`.
About	Classical Wiener space, Gaussian measure
Uses	Classical Wiener space, Wiener process, Gaussian measure, Pushforward measure, Absolutely continuous measure, Mutually singular measures, Radon–Nikodym derivative
Same as	Wikidata Q5026435 Wikipedia Cameron–Martin theorem
Changed	2 November 2026

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