Abstract: A Stochastic Exchange Model (SEM) is a Markov process in which a number of particles (or agents) repeatedly exchange a conserved quantity (discrete or continuous), such as energy or money, according to a random dynamics. We will imagine the particles as the vertices of an undirected, finite, connected graph and equip each edge with a Poisson clock: when the edge connecting x and y rings, the associated particles perform an exchange of their current amount of energy, which might be deterministic or random, depending on the model.
The simplest continuous model one can imagine is the one in which, whenever an exchange occurs, the two particles split their total energy deterministically in two halves. This model usually goes under the name of Averaging process (AVG), and has been studied extensively in recent years. In this case, regardless of the underlying graph, in the long run the system will converge to a flat configuration in which all particles carry the same amount of energy.
It is also natural to consider variants in which the exchange mechanism is itself random. A well-known instance of this class of models is the so-called Kipnis-Marchioro-Presutti model (KMP), in which, at an exchange time, the total amount of energy at the two particles is divided randomly according to a uniform distribution. In contrast to the AVG, the stationary distribution of such a dynamics is not singular, being in fact the uniform distribution on the energy simplex.
Of course, even though the graph geometry does not affect the equilibrium of these dynamics, it can have a major impact of the time the process needs to approach such a stationary state.
In this talk I will survey recent results on the convergence to equilibrium of general SEMs. I will emphasize, on the one hand, the effect of the underlying graph geometry on the convergence time of AVG and, on the other hand, the analysis of the mean-field case (i.e., the case in which the underlying graph is complete), where AVG, KMP, and several other SEMs can be shown to exhibit a universal out-of-equilibrium behavior when convergence is measured in a suitable Wasserstein metric.
The talk is based on joint work with Pietro Caputo and Federico Sau