The Gaussian space of BM

In this Chapter, a number of linear transformations of the Gaussian space associated to a linear Brownian motion (B _t , t ≥ 0) are studied. Recall that this Gaussian space is precisely equal to the first Wiener chaos of B, that is:

$$\Gamma (B) =^{\!\!\!\!\!\!\!^{{\rm def}}} \bigg\lbrace B^f \equiv \int\limits^{\infty}_0 f(s) dB_s, f \in L^2 (IR_+, ds)\bigg\rbrace$$

In fact, the properties of the transformations being studied may be deduced from corresponding properties of associated transformations of L² (IR₊, ds), thanks to the Hilbert spaces isomorphism:

$$B^f \leftrightarrow f$$

between Γ(B) and L² (IR₊, ds), which is expressed by the identity:

$$E \bigg\lbrack {(B^f)}^2\bigg\rbrack = \int\limits^{\infty}_0 dt \;f^2(t)$$

((1.1))

This chapter may be considered as a warm-up, and is intended to show that some interesting properties of Brownian motion may be deduced easily from the covariance identity (1.1).