A Scaling Method for Stochastic Process

In this work, we use the ideas of scaling to investigate stochastic process for asymptotic times, we play particular attention to the phenomenon of anomalous diffusion. The combination of method of complex variables with scaling concepts allows us to investigate the mechanism of diffusion as well for intermediates times. We generalized the concept of the diffusion exponent to include other than the asymptotic power-law behaviour. A method is proposed to obtain the diffusion coefficient analytically through the introduction of a time scaling factor lambda. We obtain also an exact expression for lambda for all kinds of diffusion. Moreover, we show that lambda is a universal parameter determined by the diffusion exponent. The results are then compared with numerical calculations and very good agreement is observed. We show the existence of two kinds of ballistic diffusion, one ergodic and another non-ergodic. The method is general and may be applied to many types of stochastic problem.