Hierarchical spatio-temporal coupling in fractional wanderings. (I) Continuous-time Weierstrass flights

The one-dimensional continuous-time Weierstrass flights (CTWF) model is considered in the framework of the nonseparable continuous-time random walks formalism (CTRW). A novel spatio-temporal coupling is introduced by assuming that in each scale the probability density for the flight and for waiting are joined. Hence, we treat the spatio-temporal relations in terms of the self-similar structure of the Weierstrass process.This (stochastic) structure is characterized by the spatial fractional dimension 1/β representing the flights and the temporal one 1/α representing the waiting. Time was assumed here as the only independent truncation range. In this work we study the asymptotic properties of the CTWF model. For example, by applying the method of steepest descents we obtained the particle propagator in the approximate scaling form, P(X,t)∼t−η(α,ptβ)/2F(ξ), where the scalingfunction F(ξ)=ξν̄(α,ptβ)exp(−const.(α,β)ξν(α,ptβ)), while the scaling variable ξ=|X|/tη(α,ptβ)/2 is large. The principal result of our analysis is that the exponents ν and ν̄ depend on more fundamental ones, α and β, what leads to a novel scaling. As a result of competition between exponents α and βan enhanced, dispersive or normal diffusion was recognized in distinction from the prediction of the separable CTRW model where the enhanced diffusion is lost and the dispersive one is strongly limited. It should be noted that we compare here partially thermalized versions of both approaches where some initial fluctuations were also included in agreement with the spirit of the theory of the renewal processes. Having the propogator we calculated, for example the particle mean-square displacement and found its novel asymptotic scaling with time for enhanced diffusion, given by ∼t1+α(2/β−1), in distinction from its diverging for β<2 within the separable CTRW model. Our version of the nonseparable CTRW approach, i.e. the CTWF model or renewed continuous-time Lévy flights (CTLF), offers a possibility to properly model the time-dependence for any fractional (critical) wandering of jump type. A similar, though mathematically more complicated analysis is also applied in Part II to the Weierstrass walks (WW) which leads to a novel version of the Lévy walks (LW) process.