  We introduce a discrete-time random walk model on a one-dimensional lattice
with a nonconstant sojourn time and prove that the discrete density converges
to a solution of a continuum diffusion equation. Our random walk model is not
Markovian due to the heterogeneity in the sojourn time, in contrast to a random
walk model with a nonconstant walk length. We derive a Markovian process by
choosing appropriate subindexes of the time-space grid points, and then show
the convergence of its discrete density through the parabolic-scale limit. We
also find the Green's function of the continuum diffusion equation and present
three Monte Carlo simulations to validate the random walk model and the
diffusion equation.
