The laws of some quadratic functionals of BM

In Chapter 1, we studied a number of properties of the Gaussian space of Brownian motion; this space may be seen as corresponding to the first level of complexity of variables which are measurable with respect to F_∞≡ σ {B _s , s ≥ 0}, where (B _s , s ≥ 0) denotes Brownian motion. Indeed, recall that N. Wiener proved that every L²(F_∞) variable X may be represented as:

$$ X = E(X) + \sum \limits_{n=1}^\infty \int\limits^{\infty}_0 dB_{t_{1}} \int\limits^{t_1}_0 dB_{t_2} \ldots \int\limits^{t_{n-1}}_0 dB_{t_n} \varphi_n (t_1,\ldots, t_n)$$

where φ _n is a deterministic Borel function which satisfies:

$$\int\limits^\infty_0 dt_1 \ldots \int\limits^{t_{n-1}}_0 dt_n\varphi_n^2 (t_1, \ldots, t_n) < \infty$$

.

In this Chapter, we shall study the laws of some of the variables X which correspond to the second level of complexity, that is: which satisfy $\varphi_n\ =\ 0$, for n ≥ 3.