Fubini's theorem for double Wiener integrals and the variance of the Brownian path (Q1178302)

The Fubini theorem for double Wiener integrals is applied to obtain identities in the law of the quadratic functionals of the Brownian motion (BMF) \(B\): \[ \int_ 0^ 1 du\left(\int_ 0^ 1 \varphi(u,s)dB_ s\right)^ 2 \buildrel {\text{ law }}\over =\int_ 0^ 1 du\left(\int_ 0^ 1 \varphi(s,u)dB_ s\right)^ 2 \] and \[ \int_ 0^ 1 du\left(Bu- \int_ 0^ 1 B_ sds\right)^ 2 \buildrel {\text{ law }}\over =\int_ 0^ 1 \tilde B_ u^ 2 du, \] where \(\tilde B\) is the Brownian bridge. For some BMF involving Legendre functions, identities in law are also established. Many concrete interesting computations related to the stochastic area are given.