It is well known that the presence of delay terms in a dynamical system can induce oscillations. However, much less is known about the effect that such memory terms have on a stochastic dynamical system. In those systems, the Markovian assumption, namely that the transition rates only depend on the current state of the system and not on its previous history, obviously wrong in many situations, is widely used because of its mathematical simplicity. After reviewing some general results on stochastic birth and death processes, I will consider in detail the case of aging, or reluctance to change state as a function of the time spent in the current state, in the voter model with noise. This is a widely used model in social and economics situations to describe transitions to consensus or synchronized behavior. While the model displays a discontinuous change of behavior from unsynchronized to consensus as a function of a parameter which depends on the tendency to act independently on the neighbors, this transition is size-dependent and disappears in the thermodynamic limit. I will show that a genuine -second order- phase transition can appear as a consequence of aging.
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