A general discrete stochastic process involving random amplification with additive external noise is analyzed theoretically and numerically. Necessary and sufficient conditions to realize steady power law fluctuations with divergent variance are clarified. The power law exponent is determined by a statistical property of amplification independent of the external noise. By introducing a nonlinear effect a stretched exponential decay appears in the power law.

We analyze statistical properties of random walkers which disappear when they meet and make offsprings by a controllable rate. Numerical results for one, two, and three dimensions and for the Sierpinski gasket are assessed in a view of the mean-field theory predictions. Universality classes are found to depend on the number of offsprings in space dimension less than 3.

We introduce a new family of aggregation models with constant interjection. In our models, the asymptotic distribution of particle mass, s, always follows a power law, P(≥s)∝$s^{\mathrm{-}\alpha }$, where (1/3≤α≤ 1) / 2 . It is clarified that this power law is realized by a balance of two effects, injection and aggregation.