Occupation times of a CTRW on a lattice with anomalous sites

We develop formalism allowing us to derive results for the probability density of the occupation time of a set of sites on a lattice of a CTRW in which the pausing-time density on the set can differ from those on the remaining lattice sites. The problem is suggested by transillumination techniques used in optical imaging. Details are given for the case of a single site on the lattice and an anomalous pausing-time density. It is shown that if the random walk is transient (i.e., in three or more dimensions when the variance of a single jump is finite) the asymptotic density for the occupation time at the special point is a negative exponential. In one dimension the random walk is recurrent. Here it is shown that the occupation time scales as . These results extend to finite numbers of anomalous points