  Recently there has been much progress in the development of stochastic models
for state reduction in quantum mechanics. In such models, the collapse of the
wave function is a physical process, governed by a nonlinear stochastic
differential equation that generalizes the Schr\"odinger equation. The present
paper considers energy-based stochastic extensions of the Schr\"odinger
equation. Most of the work carried out hitherto in this area has been concerned
with models where the process driving the stochastic dynamics of the quantum
state is Brownian motion. Here, the Brownian framework is broadened to a wider
class of models where the noise process is of the L\'evy type, admitting
stationary and independent increments. The properties of such models are
different from those of Brownian reduction models. In particular, for L\'evy
models the decoherence rate depends on the overall scale of the energy. Thus,
in L\'evy reduction models, a macroscopic quantum system will spontaneously
collapse to an eigenstate even if the energy level gaps are small.
