  We study the quantum dynamics generated by a non-Hermitian Hamiltonian
subject to stochastic perturbations in its anti-Hermitian part, describing
fluctuating gains and losses. The master equation governing the noise-average
dynamics describes a new form of dephasing. We characterize the resulting state
evolution and analyze its purity. The novel properties of such dynamics are
illustrated in a stochastic dissipative qubit. Our analytical results show that
adding noise allows for a rich control of the dynamics, with a greater
diversity of steady states and the possibility of state purification.
