Chaoticity of the stationary distribution of rank-based interacting diffusions

The mean-field limit of systems of rank-based interacting diffusions is known to be described by a nonlinear diffusion process. We obtain a similar description at the level of stationary distributions. Our proof is based on explicit expressions for the Laplace transforms of these stationary distributions and yields convergence of the marginal distributions in Wasserstein distances of all orders. We highlight the consequences of this result on the study of rank-based models of equity markets, such as the Atlas model

Sampling the Fermi statistics and other conditional product measures

Through a Metropolis-like algorithm with single step computational cost of order one, we build a Markov chain that relaxes to the canonical Fermi statistics for k non-interacting particles among m energy levels. Uniformly over the temperature as well as the energy values and degeneracies of the energy levels we give an explicit upper bound with leading term km(ln k) for the mixing time of the dynamics. We obtain such construction and upper bound as a special case of a general result on (non-homogeneous) products of ultra log-concave measures (like binomial or Poisson laws) with a global constraint. As a consequence of this general result we also obtain a disorder-independent upper bound on the mixing time of a simple exclusion process on the complete graph with site disorder. This general result is based on an elementary coupling argument and extended to (non-homogeneous) products of log-concave measures.Comment: 21 page

Optimal convergence rate of the multitype sticky particle approximation of one-dimensional diagonal hyperbolic systems with monotonic initial data

International audienceBrenier and Grenier [SIAM J. Numer. Anal., 1998] proved that sticky particle dynamics with a large number of particles allow to approximate the entropy solution to scalar one-dimensional conservation laws with monotonic initial data. In [arXiv:1501.01498], we introduced a multitype version of this dynamics and proved that the associated empirical cumulative distribution functions converge to the viscosity solution, in the sense of Bianchini and Bres-san [Ann. of Math. (2), 2005], of one-dimensional diagonal hyperbolic systems with monotonic initial data of arbitrary finite variation. In the present paper, we analyse the L 1 error of this approximation procedure, by splitting it into the discretisation error of the initial data and the non-entropicity error induced by the evolution of the particle system. We prove that the error at time t is bounded from above by a term of order (1 + t)/n, where n denotes the number of particles, and give an example showing that this rate is optimal. We last analyse the additional error introduced when replacing the multitype sticky particle dynamics by an iterative scheme based on the typewise sticky particle dynamics, and illustrate the convergence of this scheme by numerical simulations

Approximation Volumes-Finis de la mesure invariante d'une loi de conservation scalaire stochastique visqueuse

We aim to give a numerical approximation of the invariant measure of a viscous scalar conservation law, one-dimensional and periodic in the space variable, and stochastically forced with a white-in-time but spatially correlated noise. The flux function is assumed to be locally Lipschitz and to have at most polynomial growth. The numerical scheme we employ discretises the SPDE according to a finite volume method in space, and a split-step backward Euler method in time. As a first result, we prove the well-posedness as well as the existence and uniqueness of an invariant measure for both the spatial semi-discretisation and the fully discrete scheme. Our main result is then the convergence of the invariant measures of the discrete approximations, as the space and time steps go to zero, towards the invariant measure of the SPDE, with respect to the second-order Wasserstein distance. A few numerical experiments are performed to illustrate these results

LONG TIME BEHAVIOR OF MARKOV PROCESSES AND BEYOND

International audienceThis note provides several recent progresses in the study of long time behavior of Markov processes. The examples presented below are related to other scientific fields as PDE's, physics or biology. The involved mathematical tools as propagation of chaos, coupling, functional inequalities, provide a good picture of the classical methods that furnish quantitative rates of convergence to equilibrium.Cet article présente plusieurs progrés récents dans l'étude du comportement en temps long de certains processus de Markov. Les exemples présentés ci-dessous sont motivés par différentes applications issues de la physique ou de la biologie. Les outils mathématiques employés, propagation du chaos, couplage, inégalités fonctionnelles, couvrent un large spectre des techniques disponibles pour obtenir des comportements en temps long quantitatifs

A multitype sticky particle construction of Wasserstein stable semigroups solving one-dimensional diagonal hyperbolic systems with large monotonic data

International audienceThis article is dedicated to the study of diagonal hyperbolic systems in one space dimension, with cumulative distribution functions, or more generally nonconstant monotonic bounded functions, as initial data. Under a uniform strict hyperbolicity assumption on the characteristic fields, we construct a multitype version of the sticky particle dynamics and obtain existence of global weak solutions by compactness. We then derive a $L^p$ stability estimate on the particle system uniform in the number of particles. This allows to construct nonlinear semigroups solving the system in the sense of Bianchini and Bressan [Ann. of Math. (2), 2005]. We also obtain that these semigroup solutions satisfy a stability estimate in Wasserstein distances of all orders, which encompasses the classical $L^1$ estimate and generalises to diagonal systems the results by Bolley, Brenier and Loeper [J. Hyperbolic Differ. Equ., 2005] in the scalar case. Our results are obtained without any smallness assumption on the variation of the data, and only require the characteristic fields to be Lipschitz continuous and the system to be uniformly strictly hyperbolic