  We apply stochastic process theory to the analysis of immigrant integration.
Using a unique and detailed data set from Spain, we study the relationship
between local immigrant density and two social and two economic immigration
quantifiers for the period 1999-2010. As opposed to the classic time-series
approach, by letting immigrant density play the role of "time", and the
quantifier the role of "space" it become possible to analyze the behavior of
the quantifiers by means of continuous time random walks. Two classes of
results are obtained. First we show that social integration quantifiers evolve
following pure diffusion law, while the evolution of economic quantifiers
exhibit ballistic dynamics. Second we make predictions of best and worst case
scenarios taking into account large local fluctuations. Our stochastic process
approach to integration lends itself to interesting forecasting scenarios
which, in the hands of policy makers, have the potential to improve political
responses to integration problems. For instance, estimating the standard
first-passage time and maximum-span walk reveals local differences in
integration performance for different immigration scenarios. Thus, by
recognizing the importance of local fluctuations around national means, this
research constitutes an important tool to assess the impact of immigration
phenomena on municipal budgets and to set up solid multi-ethnic plans at the
municipal level as immigration pressure build.
