The Ising system with a small fraction of random long-range interactions is the simplest example of small-world phenomena in physics. Considering the latter both in an annealed and in a quenched state we conclude that: (a) the existence of random long-range interactions leads to a phase transition in the one-dimensional case and (b) there is a minimal average number p of these interactions per site (p < 1 in the annealed state, and p 1 in the quenched state) needed for the appearance of the phase transition. Note that the average number of these bonds, pN/2, is much smaller than the total number of bonds, N 2 /2.

Exact expressions have been found for the first two moments and the correlation function for an overdamped linear system subject to an external periodic field as well as to multiplicative and additive noise. Stochastic resonance is absent for Gaussian white noise. However, when the multiplicative noise has the form of an asymmetric dichotomous noise, the signal-to-noise ratio (SNR) becomes a nonmonotonic function of the correlation time and the asymmetry of noise. Moreover, the SNR turns out to be a nonmonotonic function of the frequency of the external field as well as strongly depending on the strength of the cross correlation between multiplicative and additive noise.

When the random force acting on a particle diffusing in an interval [0,L] and subjected to a constant external force is a Gaussian white noise, the "Brownian" mean-squared displacement is described by the seminal relation =2Dt(gamma) with gamma=1. However, for more complicated random forces the diffusion may be slower (gamma<1, "subdiffusion") or faster (gamma>1, "superdiffusion") than the "normal" diffusion. For both these cases we calculated the mean free passage time (MFPT)-the time needed to reach one of the traps at boundaries. The simple formulas for the different diffusive regimes are compared quantitatively for the simplest case of the absence of an external field and for an initial position in the middle of the interval. It turns out that the MFPT's for anomalous diffusion can be both larger or smaller than that for normal diffusion depending on the values of the length of the interval and the diffusion coefficient. Moreover, the MFPT can show nonmonotonic changes with the degree of departure from normal diffusion.

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