  We investigate the dynamics of a quantum system coupled linearly to Gaussian
white noise using functional methods. By performing the integration over the
noisy field in the evolution operator, we get an equivalent non-Hermitian
Hamiltonian, which evolves the quantum state with a dissipative dynamics. We
also show that if the integration over the noisy field is done for the time
evolution of the density matrix, a gain contribution from the fluctuations, can
be accessed in addition to the loss one from the non-hermitian Hamiltonian
dynamics. We illustrate our study by computing analytically the effective
non-Hermitian Hamiltonian, which we found to be the complex frequency harmonic
oscillator, with a known evolution operator. It leads to space and time
localisation, a common feature of noisy quantum systems in general
applications.
