We consider the survival probability f (t) of a random walk with a constant hopping rate w on a host lattice of fractal dimension d and spectral dimension ds ≤ 2, with spatially correlated traps. The traps form a sublattice with fractal dimension da < d and are characterized by the absorption rate wa which may be finite (imperfect traps) or infinite (perfect traps). Initial coordinates are chosen randomly at or within a fixed distance of a trap. For weakly absorbing traps (wa ≪ w), we find that f (t) can be closely approximated by a stretched exponential function over the initial stage of relaxation, with stretching exponent α = 1 − (d − da)/dw, where dw is the random walk dimension of the host lattice. At the end of this initial stage there occurs a crossover to power law kinetics f (t) ∼ t −α with the same exponent α as for the stretched exponential regime. For strong absorption wa w, including the limit of perfect traps wa → ∞, the stretched exponential regime is absent and the decay of f (t) follows, after a short transient, the aforementioned power law for all times.

We consider random walks on comb-and brush-like graphs consisting of a base (of fractal dimension D) decorated with attached side-groups. The graphs are also characterized by the fractal dimension Da of a set of anchor points where side-groups are attached to the base. Two types of graphs are considered. Graphs of the first type are uniform in the sense that anchor points are distributed periodically over the base, and thus form a subset of the base with dimension Da = D. Graphs of the second type are decorated with side-groups in a regular yet non-uniform way: the set of anchor points has fractal dimension smaller than that of the base, Da < D. For uniform graphs, a qualitative method for evaluating the sub-diffusion exponent suggested by Forte et al. for combs (D = 1) is extended for brushes (D > 1) and numerically tested for the Sierpinski brush (with the base and anchor set built on the same Sierpinski gasket). As an example of nonuniform graphs we consider the Cantor comb composed of a one-dimensional base and side-groups, the latter attached to the former at anchor points forming the Cantor set. A peculiar feature of this and other nonuniform systems is a long-lived regime of super-diffusive transport when side-groups are of a finite size.

The momentum or velocity autocorrelation function C(t) for a tagged oscillator in a finite harmonic system decays like that of an infinite system for short times, but exhibits erratic behavior at longer time scales. We introduce the autocorrelation function of the long-time noisy tail of C(t) ("a correlation of the correlation"), which characterizes the distribution of recurrence times. Remarkably, for harmonic systems with same-mass particles this secondary correlation may coincide with the primary correlation C(t) (when both functions are normalized) either exactly, or over a significant initial time interval. When the tagged particle is heavier than the rest, the equality does not hold, correlations show nonrandom long-time scale pattern, and higher-order correlations converge to the lowest normal mode.