Coupled transport in a linear-stochastic Schrödinger equation

I study heat and norm transport in a one-dimensional lattice of linear Schrödinger oscillators with conservative stochastic perturbations. Its equilibrium properties are the same of the discrete nonlinear Schrödinger equation in the limit of vanishing nonlinearity. When attached to external classical reservoirs that impose nonequilibrium conditions, the chain displays diffusive transport, with finite Onsager coefficients in the thermodynamic limit and a finite Seebeck coefficient.