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We consider a multichannel wire with a disordered region of length $L$ and areflecting boundary. The reflection of a wave of frequency $\omega$ isdescribed by the scattering matrix $\mathcal{S}(\omega)$, encoding theprobability amplitudes to be scattered from one channel to another. TheWigner-Smith time delay matrix $\mathcal{Q}=-\mathrm{i}\,\mathcal{S}^\dagger\partial_\omega\mathcal{S}$ is another important matrixencoding temporal aspects of the scattering process. In order to study itsstatistical properties, we split the scattering matrix in terms of two unitarymatrices, $\mathcal{S}=\mathrm{e}^{2\mathrm{i}kL}\mathcal{U}_L\mathcal{U}_R$(with $\mathcal{U}_L=\mathcal{U}_R^\mathrm{T}$ in the presence of TRS), andintroduce a novel symmetrisation procedure for the Wigner-Smith matrix:$\widetilde{\mathcal{Q}} =\mathcal{U}_R\,\mathcal{Q}\,\mathcal{U}_R^\dagger =(2L/v)\,\mathbf{1}_N-\mathrm{i}\,\mathcal{U}_L^\dagger\partial_\omega\big(\mathcal{U}_L\mathcal{U}_R\big)\,\mathcal{U}_R^\dagger$,where $k$ is the wave vector and $v$ the group velocity. We demonstrate that$\widetilde{\mathcal{Q}}$ can be expressed under the form of an exponentialfunctional of a matrix Brownian motion. For semi-infinite wires, $L\to\infty$,using a matricial extension of the Dufresne identity, we recoverstraightforwardly the joint distribution for $\mathcal{Q}$'s eigenvalues ofBrouwer and Beenakker [Physica E 9 (2001) p. 463]. For finite length $L$, theexponential functional representation is used to calculate the first moments$\langle\mathrm{tr}(\mathcal{Q})\rangle$,$\langle\mathrm{tr}(\mathcal{Q}^2)\rangle$ and$\langle\big[\mathrm{tr}(\mathcal{Q})\big]^2\rangle$. Finally we derive apartial differential equation for the resolvent$g(z;L)=\lim_{N\to\infty}(1/N)\,\mathrm{tr}\big\{\big( z\,\mathbf{1}_N -N\,\mathcal{Q}\big)^{-1}\big\}$ in the large $N$ limit.

Wigner-Smith matrix, exponential functional of the matrix Brownian motion and matrix Dufresne identity