Exponential functional of the matrix Brownian motion: matrix Dufresne identity and Wigner-Smith time delay matrix
Christophe Texier (LPTMS)
Hybrid seminar: onsite + zoom.
Zoom link: https://cnrs.zoom.us/j/91529089336?pwd=SlZYYnRNYjJBNjFHMVZ6YlA3WWl5dz09
Meeting ID: 915 2908 9336
Exponential functionals of the Brownian motion appear in many different contexts (classical diffusion in random media, quantum scattering, finance,…). I will discuss a recent generalization to the case of matrix Brownian motion. This problem has a natural motivation within the study of quantum scattering on a disordered wire with several conducting channels. I will show that the Wigner-Smith time delay matrix, a fundamental matrix in quantum scattering encoding several characteristic time scales, can be represented as an exponential functional of the matrix BM.
The Dufresne identity is an interesting identity in law for exponential functionals of the BM, providing the distribution of a perpetuity in risk theory. Its matrix generalization can be related to the distribution of the Wigner-Smith matrix for a semi-infinite disordered multichannel wire.
Aurélien Grabsch & Christophe Texier, Wigner-Smith matrix, exponential functional of the matrix Brownian motion and matrix Dufresne identity, J. Phys. A: Math. Theor. 53, 425003 (2020)