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We investigate the dynamics of a particle executing a general Continuous TimeRandom Walk (CTRW) in three dimensions under the influence of arbitrarytime-varying external fields. Contrary to the general approach in recent works,our method invokes neither the Fractional Fokker-Planck equation (FFPE) nor theStochastic Langevin Equation (SLE). Rather, we use rigorous probabilityarguments to derive the general expression for moments of all orders of theposition probability density of the random walker for arbitrary waiting timedensity and jump probability density. Closed form expression for the positionprobability density is derived for the memoryless condition. For the specialcase of CTRW on a one-dimensional lattice with nearest neighbour jumps, ourequations confirm the phenomena of "death of linear response" and"field-induced dispersion" for sub-diffusion pointed out in [I. M. Sokolov andJ. Klafter, Phys. Rev. Lett. {\bf 97}, 140602 (2006)]. However, our analysisproduces additional terms in the expressions for higher moments, which havenon-trivial consequences. We show that the disappearance of these terms resultfrom the approximation involved in taking the continuum limit to derive thegeneralized Fokker-Planck equation. This establishes the incompleteness of theFFPE formulation, especially in predicting the higher moments. We also discusshow different predictions of the model alter if we allow jumps beyond nearestneighbours and possible circumstances where this becomes relevant.

Continuous Time Random Walk with time-dependent jump probability : A Direct Probabilistic Approach