288 research outputs found
Some Fluctuation Results for Weakly Interacting Multi-type Particle System
A collection of -diffusing interacting particles where each particle
belongs to one of different populations is considered. Evolution equation
for a particle from population depends on the empirical measures of
particle states corresponding to the various populations and the form of this
dependence may change from one population to another. In addition, the drift
coefficients in the particle evolution equations may depend on a factor that is
common to all particles and which is described through the solution of a
stochastic differential equation coupled, through the empirical measures, with
the -particle dynamics. We are interested in the asymptotic behavior as
. Although the full system is not exchangeable, particles in the
same population have an exchangeable distribution. Using this structure, one
can prove using standard techniques a law of large numbers result and a
propagation of chaos property. In the current work we study fluctuations about
the law of large number limit. For the case where the common factor is absent
the limit is given in terms of a Gaussian field whereas in the presence of a
common factor it is characterized through a mixture of Gaussian distributions.
We also obtain, as a corollary, new fluctuation results for disjoint
sub-families of single type particle systems, i.e. when . Finally, we
establish limit theorems for multi-type statistics of such weakly interacting
particles, given in terms of multiple Wiener integrals.Comment: 47 page
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
Supermarket Model on Graphs
We consider a variation of the supermarket model in which the servers can
communicate with their neighbors and where the neighborhood relationships are
described in terms of a suitable graph. Tasks with unit-exponential service
time distributions arrive at each vertex as independent Poisson processes with
rate , and each task is irrevocably assigned to the shortest queue
among the one it first appears and its randomly selected neighbors. This
model has been extensively studied when the underlying graph is a clique in
which case it reduces to the well known power-of- scheme. In particular,
results of Mitzenmacher (1996) and Vvedenskaya et al. (1996) show that as the
size of the clique gets large, the occupancy process associated with the
queue-lengths at the various servers converges to a deterministic limit
described by an infinite system of ordinary differential equations (ODE). In
this work, we consider settings where the underlying graph need not be a clique
and is allowed to be suitably sparse. We show that if the minimum degree
approaches infinity (however slowly) as the number of servers approaches
infinity, and the ratio between the maximum degree and the minimum degree in
each connected component approaches 1 uniformly, the occupancy process
converges to the same system of ODE as the classical supermarket model. In
particular, the asymptotic behavior of the occupancy process is insensitive to
the precise network topology. We also study the case where the graph sequence
is random, with the -th graph given as an Erd\H{o}s-R\'enyi random graph on
vertices with average degree . Annealed convergence of the occupancy
process to the same deterministic limit is established under the condition
, and under a stronger condition ,
convergence (in probability) is shown for almost every realization of the
random graph.Comment: 32 page
Moderate deviation principles for weakly interacting particle systems
Moderate deviation principles for empirical measure processes associated with weakly interacting Markov processes are established. Two families of models are considered: the first corresponds to a system of interacting diffusions whereas the second describes a collection of pure jump Markov processes with a countable state space. For both cases the moderate deviation principle is formulated in terms of a large deviation principle (LDP), with an appropriate speed function, for suitably centered and normalized empirical measure processes. For the first family of models the LDP is established in the path space of an appropriate Schwartz distribution space whereas for the second family the LDP is proved in the space of (the Hilbert space of square summable sequences)-valued paths. Proofs rely on certain variational representations for exponential functionals of Brownian motions and Poisson random measures
Graphon mean field systems
We consider heterogeneously interacting diffusive particle systems and their
large population limit. The interaction is of mean field type with weights
characterized by an underlying graphon. A law of large numbers result is
established as the system size increases and the underlying graphons converge.
The limit is given by a graphon mean field system consisting of independent but
heterogeneous nonlinear diffusions whose probability distributions are fully
coupled. Well-posedness, continuity and stability of such systems are provided.
We also consider a not-so-dense analogue of the finite particle system,
obtained by percolation with vanishing rates and suitable scaling of
interactions. A law of large numbers result is proved for the convergence of
such systems to the corresponding graphon mean field system.Comment: 31 pages. Continuity assumptions on graphons are relaxed; system
coefficients are allowed to have linear growt
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