Exotic stochastic processes are shown to emerge in the quantum evolution of complex systems. Using influence function techniques, we consider the dynamics of a system coupled to a chaotic subsystem described through random matrix theory. We find that the reduced density matrix can display dynamics given by Lévy stable laws. The classical limit of these dynamics can be related to fractional kinetic equations. In particular we derive a fractional extension of Kramers equation.PACS numbers: 05.40.+j, 02.50.-r, 05.30.-d, 05.45.+b Whether one studies deterministic Hamiltonian or dissipative systems, one finds that transport in chaotic systems often resembles some type of stochastic process. The dynamics of such systems leads to a rich spectrum of behaviors, from enhanced diffusion such as tracer diffusion in flows and turbulent diffusion in the atmosphere, to dispersive diffusion [1]. Much effort has been spent in recent years to understand such classical stochastic processes in chaotic systems, leading to the development of approaches ranging from fractional kinetic equations [2-4], Lévy flights [5] to random walks in random environments [5,6] and stochastic webs [7]. One of the common features to all of these is the use of Lévy stable laws [8]. It was shown by Lévy [9], in studies of extensions of the central limit theorem, that a continuous class of nongaussian processes satisfy the same fundamental equation that gives rise to the theory of gaussian processes, namely the Chapman-Kolmogorov equation for the conditional probability P (q, q ′ , t):The standard solution, P (q, q ′ , t) = exp(−(q − q ′ ) 2 /4Dt)/(4πDt) 3/2 , gives rise to the gaussian processes and the usual form of the Fokker-Planck equation. The general solutions of Lévy provide a way to generalize Brownian motion. The non-gaussian processes which satisfy (1) are known as Lévy stable laws, and have the form:where 0 < α ≤ 2 and A ∝ t. The case α = 2 corresponds to gaussian processes. The Lévy distributions L A α (q) satisfy the scaling relation:where for A = 1 we drop the superscript: L 1 α (x) = L α (x). For α < 2, these distributions are characterized by infinite second moments, as one can easily infer from the asymptotic behavior for q → ±∞ [5],These non-gaussian processes can be related to anomalous transport in a variety of (classical) physical systems [6], as well as to classically chaotic systems. We have recently shown that turbulent diffusion can also arise in the time evolution of complex quantum systems [10]. Here we find that a general form of quantum chaotic backgrounds can give rise to quantum evolution characterized by Lévy distributions. Further, we can connect, in the semi-classical limit, such processes to fractional kinetic theory, which was initially introduced as a phenomenological approach to classical anomalous diffusion.We would like to study the problem of a particle coupled to a chaotic environment, quantum mechanically. It has been realized in recent years that the quantum counterpart of chaos is random matrix theory. For ...

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