V-Langevin equations, continuous time random walks and fractional diffusion

The following question is addressed: under what conditions can a strange diffusive process, defined by a semi-dynamical V-Langevin equation or its associated hybrid kinetic equation (HKE), be described by an equivalent purely stochastic process, defined by a continuous time random walk (CTRW) or by a fractional differential equation (FDE)? More specifically, does there exist a class of V-Langevin equations with long-range (algebraic) velocity temporal correlation, that leads to a time-fractional superdiffusive process? The answer is always affirmative in one dimension. It is always negative in two dimensions: any algebraically decaying temporal velocity correlation (with a Gaussian spatial correlation) produces a normal diffusive process. General conditions relating the diffusive nature of the process to the temporal exponent of the Lagrangian velocity correlation (in Corrsin approximation) are derived. It is shown that a bifurcation occurs as the latter parameter is varied. Above that bifurcation value the process is always diffusive.