Probability evolution for complex multi-linear non-local interactions

Irene Gamba
University of Texas at Austin

We present a generalized formulation of kinetic non-local models of collisional type that cover a large class of global energy dissipative
phenomena, such as those of inelastic collisions, mixtures and slowdown cooling processes, economics and social dynamics; in the setting of
multiplicatively interactive stochastic processes.
The working framework lays in the space of characteristic functions of probabilities measures for a class of equations of non-local multi linear form, where is possible to recover the longtime dynamics as those of (W)_2 metric evolution to stable or dynamically stable states self-similar) for finite energy; but also the technique extends
them to cases of infinity energy initial data, where the classical definition of W_2 metric does not apply. One of the consequences is the existence and characterization of the dynamically
stable (self-similar) states in probability space which can not have all moments bounded and even may admit singularities at the origin, while remaining integrable (as in examples of limit mixture models for a slow-down process). Part of this work has been in collaboration with A. Bobylev and C. Cergignani

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