The Fokker-Planck equation has been very useful for studying dynamic behavior of stochastic differential equations driven by Gaussian 1 noises. However, there are both theoretical and empirical reasons to consider similar equations driven by strongly non-Gaussian noises. In particular, they yield strongly non-Gaussian anomalous diffusion which seems to be relevant in different domains of Physics. We therefore derive in this paper a Fractional Fokker-Planck equation for the probability distribution of particles whose motion is governed by a nonlinear Langevin-type equation, which is driven by a Levy-stable noise rather than a Gaussian. We obtain in fact a general result for a Markovian forcing. We also discuss the existence and uniqueness of the solution of the Fractional Fokker-Planck equation.

J. Dynamics and Diff. Eqns 2004, in press. Invariant manifolds are fundamental tools for describing and understanding nonlinear dynamics. In this paper, we present a theory of stable and unstable manifolds for infinite dimensional random dynamical systems generated by a class of stochastic partial differential equations. We first show the existence of Lipschitz continuous stable and unstable manifolds by the Lyapunov-Perron's method. Then, we prove the smoothness of these invariant manifolds.

A Boussinesq model for the Bénard convection under random influences is considered as a system of stochastic partial differential equations. This is a coupled system of stochastic Navier-Stokes equations and the transport equation for temperature. Large deviations are proved, using a weak convergence approach based on a variational representation of functionals of infinite-dimensional Brownian motion.2000 Mathematics Subject Classification. Primary 60H15, 60F10; Secondary 35R60, 76D05, 76R05 .

The mean first exit time and escape probability are utilized to quantify dynamical behaviors of stochastic differential equations with non-Gaussian α-stable type Lévy motions. An efficient and accurate numerical scheme is developed and validated for computing the mean exit time and escape probability from the governing differential-integral equation. An asymptotic solution for the mean exit time is given when the pure jump measure in the Lévy motion is small. From both the analytical and numerical results, it is observed that the mean exit time depends strongly on the domain size and the value of α in the α-stable Lévy jump measure. The mean exit time and escape probability could become discontinuous at the boundary of the domain, when the value of α is in (0, 1). Introduction.Random fluctuations in complex systems in engineering and science are often non-Gaussian [33,12,11]. For instance, it has been argued that diffusion by geophysical turbulence [29] corresponds, loosely speaking, to a series of "pauses," when the particle is trapped by a coherent structure, and "flights" or "jumps" or other extreme events, when the particle moves in the jet flow. Paleoclimatic data [13] also indicate such irregular processes.Lévy motions are thought to be appropriate models for non-Gaussian processes with jumps [26]. Recall that a Lévy motion L(t), or L t , is a stochastic process with stationary and independent increments. That is, for any s, t with 0 ≤ s < t, the distribution of L t − L s only depends on t − s, and for any 0 ≤ t 0 < t 1 < · · · < t n , L ti − L ti−1 , i = 1, . . . , n, are independent. Without loss of generality, we may assume that the sample paths of L t are almost surely right continuous with left limits.This generalizes the Brownian motion B(t), which satisfies all three conditions. But additionally, (i) almost all sample paths of the Brownian motion are continuous in time in the usual sense and (ii) Brownian motion's increments are Gaussian distributed.