Deformed random walk: Suppression of randomness and inhomogeneous diffusion

We study a generalization of the random walk (RW) based on a deformed translation of the unitary step, inherited by the q algebra, a mathematical structure underlying nonextensive statistics. The RW with deformed step implies an associated deformed random walk (DRW) provided with a deformed Pascal triangle along with an inhomogeneous diffusion. The paths of the RW in deformed space are divergent, while those corresponding to the DRW converge to a fixed point. Standard random walk is recovered for q→1 and a suppression of randomness is manifested for the DRW with -1<γ_{q}<1 and γ_{q}=1-q. The passage to the continuum of the master equation associated to the DRW led to a van Kampen inhomogeneous diffusion equation when the mobility and the temperature are proportional to 1+γ_{q}x, and provided with an exponential hyperdiffusion that exhibits a localization of the particle at x=-1/γ_{q} consistent with the fixed point of the DRW. Complementarily, a comparison with the Plastino-Plastino Fokker-Planck equation is discussed. The two-dimensional case is also studied, by obtaining a 2D deformed random walk and its associated deformed 2D Fokker-Planck equation, which give place to a convergence of the 2D paths for -1<γ_{q_{1}},γ_{q_{2}}<1 and a diffusion with inhomogeneities controlled by two deformation parameters γ_{q_{1}},γ_{q_{2}} in the directions x and y. In both the one-dimensional and the two-dimensional cases, the transformation γ_{q}→-γ_{q} implies a change of sign of the corresponding limits of the random walk paths, as a property of the deformation employed.