Stochastic equations constitute a major ingredient in many branches of science, from physics
to biology and engineering. Not surprisingly, they appear in many quantitative studies of complex
systems. In particular, this type of equation is useful for understanding the dynamics of urban population which
will be the subject of this talk. Empirically, the population of cities follows a seemingly universal law - called Zipf’s law -
which was discovered about a century ago and states that when sorted in decreasing order, the population
of a city varies as the inverse of its rank. In addition, the ranks of cities follow a turbulent dynamics: some cities rise
while other fall and disappear. Both these aspects - Zipf’s law (and deviations around this law), and the turbulent
dynamics of ranks - need to be explained by the same theoretical framework and it is natural to look for the equation that
governs the evolution of urban populations. In this talk I will review the main theoretical attempts based on stochastic equations
to describe these empirical facts.
