Random Walk on Z with One-Point Inhomogeneity

We consider a locally non-homogeneous discrete-time random walk on Z, with the non-homogeneous perturbation concentrated at the origin. We assume exponential decay of the probabilities and symmetry for the homogeneous term. Combining probabilistic and analytic methods, we find an explicit expression for the asymptotic behavior of the probabilities P(X-t = x vertical bar X-0 = 0), as t -> infinity, which holds uniformly for x = o(t(3/4)). We also discuss the probabilistic interpretation of the results.