CameronMartinTheorem - crowlogic/arb4j GitHub Wiki

The Cameron-Martin theorem is a fundamental result in the theory of Gaussian measures on infinite-dimensional spaces, especially in the context of Wiener measure, which is a measure on the space of continuous paths associated with Brownian motion. This theorem provides necessary and sufficient conditions for a path to be absolutely continuous with respect to the Wiener measure.

To state the theorem, it's essential to first introduce some concepts and notation:

  1. Wiener Measure: Denoted by $\mu$, it's a probability measure on the space of continuous functions $C([0,1]; \mathbb{R}^d)$ such that $\omega(0) = 0$. Here, $\omega$ represents a path in the space, and $d$ is the dimension.

  2. Cameron-Martin Space: The Cameron-Martin space $H$ is a Hilbert space of functions $h: [0,1] \rightarrow \mathbb{R}^d$ that have a version which is absolutely continuous and satisfy:

$$ h(t) = \int_0^t h'(s) ds $$ with $h'$ square integrable over [0,1]. The inner product on this space is given by: $$ \langle h, k \rangle_H = \int_0^1 h'(t) k'(t) dt $$

Cameron-Martin Theorem: Let $\mu$ be the Wiener measure on $C([0,1]; \mathbb{R}^d)$ and let $h$ be a path in the Cameron-Martin space $H$. Then the shifted measure $\mu_h$ given by:

$$ \mu_h(A) = \mu({ \omega + h : \omega \in A })$$

is absolutely continuous with respect to $\mu$, and the Radon-Nikodym derivative is:

$$ \frac{d\mu_h}{d\mu}(\omega) = \exp\left( -\frac{1}{2} | h |_H^2 + \langle \omega, h \rangle_H \right)$$

where $\langle \cdot, \cdot \rangle_H$ is the inner product in the Cameron-Martin space.

The theorem also states that paths outside the Cameron-Martin space will give rise to measures which are singular with respect to the Wiener measure.

This theorem is of central importance in many areas, particularly in the study of stochastic calculus and the theory of stochastic differential equations. The result tells us about the "geometry" of the space of Brownian paths, in the sense that it tells us which paths can be reached from others by an absolutely continuous change of measure.