Theses Doctoral

Topics in large-scale limits of interacting systems: games with common noise, quantitative propagation of chaos and networks.

Le Flem, Luc

The study of large interacting particle systems has broad applications in many scientific fields such as statistical physics, social sciences, bio-sciences and finance. This thesis proposes to study various theoretical problems in order to improve our understanding of the asymptotics of such systems, in the context of stochastic diffusions. Namely, we present diverse circumstances where the mean field limit becomes a good approximation for these systems. This manuscript contains the result of three different projects.

The first one shows the optimal rate of propagation of chaos, uniformly in time, for interacting diffusions. This project pursues the recent trend of the study of quantitative, as opposed to qualitative, propagation of chaos. Most previous approaches use synchronous coupling arguments. We propose a different approach, based on the analysis of the relative entropy between any π‘˜β€“ particle system, and its limiting product measure, through the BBGKY hierarchy. The main applications of our results are for models with convex potentials and models on the torus with small interactions.

The second project studies the large population limit for asymmetric interacting diffusions, where the interactions are governed by an underlying network. We introduce a specific kind of McKean-Vlasov equation that we show captures the asymptotics of the interacting particle system. Our approach relies on weak compactness arguments and requires only mild assumptions on the coefficients of the diffusions. This brings the qualitative limit theory for heterogeneous interacting diffusions to the same level of generality that is known for the homogeneous case. On the graph, we require convergence in cut norm, and, in the case of unbounded coefficients, the additional assumption that the row and column average of the interaction matrix are uniformly bounded. In particular, our approach covers the example of Erdos-Renyi graphs in the case 𝑁𝑃𝑛 β†’ ∞ for bounded coefficients and the case lim inf𝑛 β†’ βˆžπ‘π‘ƒπ‘› / log(𝑛) > 0 for unbounded coefficients.

Our last project shifts the focus from interacting particles to games. We place ourselves back in the symmetric case but with the additional presence of a common noise, and define a notion of weak mean field equilibria. We show that this notion captures all sequential limit points, as 𝑛 β†’ ∞, of closed-loop approximate equilibria from the 𝑛-player game. We also show that every weak mean field equilibrium can be used to construct closed-loop approximate equilibria at the level of the 𝑛-player game, as long as the latter are formulated over a broader class of closed-loop strategies which may depend on an additional common signal.


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More About This Work

Academic Units
Industrial Engineering and Operations Research
Thesis Advisors
Lacker, Daniel H.
Ph.D., Columbia University
Published Here
February 15, 2023