99 research outputs found
Translation invariant mean field games with common noise
This note highlights a special class of mean field games in which the
coefficients satisfy a convolution-type structural condition. A mean field game
of this type with common noise is related to a certain mean field game without
common noise by a simple transformation, which permits a tractable construction
of a solution of the problem with common noise from a solution of the problem
without
A probabilistic weak formulation of mean field games and applications
Mean field games are studied by means of the weak formulation of stochastic
optimal control. This approach allows the mean field interactions to enter
through both state and control processes and take a form which is general
enough to include rank and nearest-neighbor effects. Moreover, the data may
depend discontinuously on the state variable, and more generally its entire
history. Existence and uniqueness results are proven, along with a procedure
for identifying and constructing distributed strategies which provide
approximate Nash equlibria for finite-player games. Our results are applied to
a new class of multi-agent price impact models and a class of flocking models
for which we prove existence of equilibria
Local weak convergence and propagation of ergodicity for sparse networks of interacting processes
We study the limiting behavior of interacting particle systems indexed by
large sparse graphs, which evolve either according to a discrete time Markov
chain or a diffusion, in which particles interact directly only with their
nearest neighbors in the graph. To encode sparsity we work in the framework of
local weak convergence of marked (random) graphs. We show that the joint law of
the particle system varies continuously with respect to local weak convergence
of the underlying graph. In addition, we show that the global empirical measure
converges to a non-random limit, whereas for a large class of graph sequences
including sparse Erd\"{o}s-R\'{e}nyi graphs and configuration models, the
empirical measure of the connected component of a uniformly random vertex
converges to a random limit. Finally, on a lattice (or more generally an
amenable Cayley graph), we show that if the initial configuration of the
particle system is a stationary ergodic random field, then so is the
configuration of particle trajectories up to any fixed time, a phenomenon we
refer to as "propagation of ergodicity". Along the way, we develop some general
results on local weak convergence of Gibbs measures in the uniqueness regime
which appear to be new.Comment: 46 pages, 1 figure. This version of the paper significantly extends
the convergence results for diffusions in v1, and includes new results on
propagation of ergodicity and discrete-time models. The complementary results
in v1 on autonomous characterization of marginal dynamics of diffusions on
trees and generalizations thereof are now presented in a separate paper
arXiv:2009.1166
Mean field games with common noise
A theory of existence and uniqueness is developed for general stochastic
differential mean field games with common noise. The concepts of strong and
weak solutions are introduced in analogy with the theory of stochastic
differential equations, and existence of weak solutions for mean field games is
shown to hold under very general assumptions. Examples and counter-examples are
provided to enlighten the underpinnings of the existence theory. Finally, an
analog of the famous result of Yamada and Watanabe is derived, and it is used
to prove existence and uniqueness of a strong solution under additional
assumptions
- …