Complex Obtuse Random Walks and their Continuous-Time Limits

We study a particular class of complex-valued random variables and their associated random walks: complex obtuse random variables. They generalize, to the complex case, the real-valued obtuse random variables introduced in Equations de structure pour des martingales vectorielles. (Seminaire de Probabilites, XXVIII, p. 256278. Lecture Notes in Math., vol. 1583. Springer, Berlin (1994)) in order to understand the structure of normal martingales in \(\mathbb {R}^N\). The extension to the complex case is motivated by Quantum Statistical Mechanics, in particular for characterizing those quantum baths acting as classical noises. The extension of obtuse random variables to the complex case is far from obvious and makes use of very interesting algebraical structures. We show that complex obtuse random variables are characterized by a 3-tensor which admits certain symmetries; we show that these symmetries are the exact 3-tensor analogue of the normal character for 2-tensors (i.e. matrices), that is, a necessary and sufficient condition for being diagonalizable in some orthonormal basis. We discuss the passage to the continuous-time limit for these random walks and show that they converge in distribution to normal martingales in \(\mathbb {C}^N\). We show that the 3-tensor associated to these normal martingales encodes their behavior, in particular the diagonalization directions of the 3-tensor indicate the directions of the space where the martingales behaves like a diffusion and those where it behaves like a Poisson process. We finally prove the convergence, in the continuous-time limit, of the corresponding multiplication operators on the canonical Fock space, with an explicit expression in terms of the associated 3-tensor again.