  Stochastic processes are shown to emerge from the time evolution of complex
quantum systems. Using parametric, banded random matrix ensembles to describe a
quantum chaotic environment, we show that the dynamical evolution of a particle
coupled to such environments displays a variety of stochastic behaviors,
ranging from turbulent diffusion to L\'evy processes and Brownian motion.
Dissipation and diffusion emerge naturally in the stochastic interpretation of
the dynamics. This approach provides a derivation of a fractional kinetic
theory in the classical limit and leads to classical L\'evy dynamics.
