Random walks with homotopic spatial inhomogeneities

Abstract: In this work we study a generalization of the standard random walk, an homotopic random walk (HRW), using a deformed translation unitary step that arises from a homotopy of the position-dependent masses associated to the Tsallis and Kaniadakis nonexensive statistics. The HRW implies an associated homotopic Fokker-Planck equation (HFPE) provided with a bi-parameterized inhomogeneous diffusion. The trajectories of the HRW exhibit convergence to a position, randomness as well as divergence, according to deformation and homotopic parameters. The HFPE obtained from associated master equation to the HRW presents the features: a) it results an special case of the van Kampen diffusion equation (5) of Ref. [N. G. van Kampen, \emph{Z. Phys. B Condensed Matter} \textbf{68}, 135 (1987)]; b) it exhibits a superdiffusion in function of deformation and homotopic parameters; c) Tsallis and Kaniadakis deformed FPE are recovered as special cases; d) a homotopic mixtured diffusion is observed; and e) it has a stationary entropic density, characterizing a inhomogeneous screening of the medium, obtained from a homotopic version of the H-Theorem.