We present Markov models for two social processes: the spread of rumors and the change in the spatial distribution of a population over time. For the spread of rumors, we present two models. The first is for the situation in which all particles are identical but one initially knows the rumor. The second is for a situation in which there are two kinds of particles: spreaders, who can spread the rumor, and ordinary particles, who only can learn the rumor. We find that the limiting distribution for the first model is the convolution of two double exponential distributions and for the second model is a double exponential distribution.The stochastic dynamics for our model of the change in the spatial distribution of a population over time include the four basic demographic processes: birth, death, migration, and immigration. We allow interaction between particles only inasmuch as the immigration rate can depend on the existing configuration of particles. We focus on the critical case of constant mean density, under the conditions of long jumps migration, immigration in which distant particles have a positive effect, or both. We prove, under these conditions, the existence of ergodic limiting behavior: the point process is stationary in space and time. Without the strong mixing due to these conditions, the population vanishes due to infinite clusterization.
