536 research outputs found
Optimal coupling for mean field limits
We review recent quantitative results on the approximation of mean field
diffusion equations by large systems of interacting particles, obtained by
optimal coupling methods. These results concern a larger range of models, more
precise senses of convergence and links with the long time behaviour of the
systems to be considered
Quantitative concentration inequalities on sample path space for mean field interaction
We consider a system of particles experiencing diffusion and mean field
interaction, and study its behaviour when the number of particles goes to
infinity. We derive non-asymptotic large deviation bounds measuring the
concentration of the empirical measure of the paths of the particles around its
limit. The method is based on a coupling argument, strong integrability
estimates on the paths in Holder norm, and some general concentration result
for the empirical measure of identically distributed independent paths
Phi-entropy inequalities for diffusion semigroups
We obtain and study new -entropy inequalities for diffusion semigroups,
with Poincar\'e or logarithmic Sobolev inequalities as particular cases. From
this study we derive the asymptotic behaviour of a large class of linear
Fokker-Plank type equations under simple conditions, widely extending previous
results. Nonlinear diffusion equations are also studied by means of these
inequalities. The criterion of D. Bakry and M. Emery appears as a
main tool in the analysis, in local or integral forms.Comment: 31 page
Concentration of measure on product spaces with applications to Markov processes.
For a stochastic process with state space some Polish space, this paper gives sufficient conditions on the initial and conditional distributions for the joint law to satisfy Gaussian concentration and transportation inequalities. In the case of Euclidean space, there are sufficient conditions for the joint law to satisfy a logarithmic Sobolev inequality. In several cases, the constants obtained are of optimal growth with respect to the number of random variables, or are independent of this number. These results extend results known for mutually independent random variables and weakly dependent random variabels under Dobrushkin--Shlosman type conditions. The paper also contains applications to Markov processes including the ARMA process
Phi-entropy inequalities and Fokker-Planck equations
We present new -entropy inequalities for diffusion semigroups under the
curvature-dimension criterion. They include the isoperimetric function of the
Gaussian measure. Applications to the long time behaviour of solutions to
Fokker-Planck equations are given
Non ultracontractive heat kernel bounds by Lyapunov conditions
Nash and Sobolev inequalities are known to be equivalent to ultracontractive
properties of heat-like Markov semigroups, hence to uniform on-diagonal bounds
on their kernel densities. In non ultracontractive settings, such bounds can
not hold, and (necessarily weaker, non uniform) bounds on the semigroups can be
derived by means of weighted Nash (or super-Poincar\'e) inequalities. The
purpose of this note is to show how to check these weighted Nash inequalities
in concrete examples, in a very simple and general manner. We also deduce
off-diagonal bounds for the Markov kernels of the semigroups, refining E. B.
Davies' original argument
Dynamics of a planar Coulomb gas
We study the long-time behavior of the dynamics of interacting planar
Brow-nian particles, confined by an external field and subject to a singular
pair repulsion. The invariant law is an exchangeable Boltzmann -- Gibbs
measure. For a special inverse temperature, it matches the Coulomb gas known as
the complex Ginibre ensemble. The difficulty comes from the interaction which
is not convex, in contrast with the case of one-dimensional log-gases
associated with the Dyson Brownian Motion. Despite the fact that the invariant
law is neither product nor log-concave, we show that the system is well-posed
for any inverse temperature and that Poincar{\'e} inequalities are available.
Moreover the second moment dynamics turns out to be a nice Cox -- Ingersoll --
Ross process in which the dependency over the number of particles leads to
identify two natural regimes related to the behavior of the noise and the speed
of the dynamics.Comment: Minor revision for Annals of Applied Probabilit
Dimensional contraction via Markov transportation distance
It is now well known that curvature conditions \`a la Bakry-Emery are
equivalent to contraction properties of the heat semigroup with respect to the
classical quadratic Wasserstein distance. However, this curvature condition may
include a dimensional correction which up to now had not induced any
strenghtening of this contraction. We first consider the simplest example of
the Euclidean heat semigroup, and prove that indeed it is so. To consider the
case of a general Markov semigroup, we introduce a new distance between
probability measures, based on the semigroup, and adapted to it. We prove that
this Markov transportation distance satisfies the same properties for a general
Markov semigroup as the Wasserstein distance does in the specific case of the
Euclidean heat semigroup, namely dimensional contraction properties and
Evolutional variational inequalities
Mean-field limit for the stochastic Vicsek model
We consider the continuous version of the Vicsek model with noise, proposed
as a model for collective behavior of individuals with a fixed speed. We
rigorously derive the kinetic mean-field partial differential equation
satisfied when the number N of particles tends to infinity, quantifying the
convergence of the law of one particle to the solution of the PDE. For this we
adapt a classical coupling argument to the present case in which both the
particle system and the PDE are defined on a surface rather than on the whole
space. As part of the study we give existence and uniqueness results for both
the particle system and the PDE
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