Coarse-graining of asymmetric discrete-time random walk on a one-dimensional lattice

Coarse-graining of an asymmetric nearest-neighbor discrete-time random walk on a one-dimensional lattice allows one to describe this random walk as biased one-dimensional diffusion. The latter is characterized by two parameters: the drift velocity and diffusivity. There is a general expression giving the drift velocity as a function of the parameters determining the random walk. However, a corresponding expression for the diffusivity is known only for the particular case where the random walk escapes from the lattice site at every time step. In this work, we generalize this result and derive an expression for the diffusivity, assuming that the random walk does not necessarily leave the site, and therefore, its mean lifetime on the site can be longer than the time step.