On the extension of an identity in law between the Brownian bridge and the variance of the Brownian motion (Q1316599)

M. Yor established in 1991 that the laws of \(\int^ 1_ 0 [B_ s- s^{-1} \int^ s_ 0 B_ u du]^ 2 ds\) and of \(\int^ 1_ 0 \widetilde{B}_ s ds\) were equal, \(B\) being standard real Brownian motion and \(\widetilde{B}\) standard real Brownian bridge, and he calculated the Laplace transform of \(\int^ 1_ 0 B_ s B_ s'ds\) (resp. \(\int^ 1_ 0 \widetilde{B}_ s \widetilde{B}_ s' ds\)), \(B'\) (resp. \(\widetilde{B}'\)) being i.i.d. with \(B\) (resp. \(\widetilde{B}\)). The author presents an extension of these results to the case where \(B'\) (resp. \(\widetilde {B}'\)) is replaced by \(\rho B+ \sqrt{1-\rho^ 2} B'\) (resp. \(\rho \widetilde{B}+ \sqrt{1- \rho^ 2} \widetilde{B}'\)), \(\rho\) being a constant. His calculation uses Mercer's \(L^ 2\)-decomposition of \(B_ s\) (resp. \(\widetilde{B}_ s\)) and diagonalization of the kernel \(s\Lambda s'\) (resp. \(s\Lambda s'- ss'\)).