Searcharxiv⌕ Search

arXiv subjects

Pierre Monmarché

Publications and source records attributed to Pierre Monmarché.

81 records · Page 5Linked to original sources
arXiv2018

Generalized $Γ$ calculus and application to interacting particles on a graph

The classical Bakry-Émery calculus is extended to study, for degenerated (non-elliptic, non-reversible, or non-diffusive) Markov processes, questions such as hypoellipticity, hypocoercivity, functional inequalities or Wasserstein contraction. In particular we obtain the optimal speed of convergence to equilibrium for any ergodic Ornstein-Uhlenbeck process, which is given by the spectral gap of the drift matrix and the size of the corresponding Jordan blocks. We also study chains of $N$ interacting overdamped particles and establish for their invariant measures log-Sobolev inequalities with constants of order $N^2$, which is optimal.

math.PR↗
arXiv2017

Ergodicity and propagation of chaos for mean field kinetic particles

The trend to equilibrium in large time is studied for a large particle system associated to a Vlasov-Fokker-Planck equation in the presence of a convex external potential, without smallness restriction on the interaction. From this are derived uniform in time propagation of chaos estimates, which themselves yield in turn an exponentially fast convergence for the semi-linear equation itself. The approach is quantitative.

math.PR↗
arXiv2017

Entropic multipliers method for langevin diffusion and weighted log sobolev inequalities

In his work about hypocercivity, Villani [18] considers in particular convergence to equilibrium for the kinetic Langevin process. While his convergence results in L 2 are given in a quite general setting, convergence in entropy requires some boundedness condition on the Hessian of the Hamiltonian. We will show here how to get rid of this assumption in the study of the hypocoercive entropic relaxation to equilibrium for the Langevin diffusion. Our method relies on a generalization to entropy of the multipliers method and an adequate functional inequality. As a byproduct, we also give tractable conditions for this functional inequality, which is a particular instance of a weighted logarithmic Sobolev inequality, to hold.

math.PR↗
arXiv2016

Piecewise deterministic simulated annealing

Given an energy potential on the Euclidian space, a piecewise deterministic Markov process is designed to sample the corresponding Gibbs measure. In dimension one an Eyring-Kramers formula is obtained for the exit time of the domain of a local minimum at low temperature, and a necessary and sufficient condition is given on the cooling schedule in a simulated annealing algorithm to ensure the process converges to the set of global minima. This condition is similar to the classical one for diffusions and involves the critical depth of the potential. In higher dimension a non optimal sufficient condition is obtained.

math.PR↗
arXiv2015

On $\mathcal H^1$ and entropic convergence for contractive PDMP

Explicit rate of convergence in variance (or more general entropies) is obtained for a class of Piecewise Deterministic Markov Processes such as the TCP process, relying on functional inequalities. A method to establish Poincaré (and more generaly Beckner) inequalities with respect to a diffusion-type energy for the invariant law of such hybrid processes is developped.

math.PR↗
arXiv2014

Hypocoercive relaxation to equilibrium for some kinetic models via a third order differential inequality

This paper deals with the study of some particular kinetic models, where the randomness acts only on the velocity variable level. Usually, the Markovian generator cannot satisfy any Poincaré's inequality. Hence, no Gronwall's lemma can easily lead to the exponential decay of Ft (the L2 norm of a test function along the semi-group). Nevertheless for the kinetic Fokker-Planck dynamics and for a piecewise deterministic evolution we show that Ft satisfies a third order differential inequality which gives an explicit rate of convergence to equilibrium.

math.PR↗
arXiv2012

Étude spectrale minutieuse de processus moins indécis que les autres

In this paper we are looking for quantitative estimates for the convergene to equilibrium of non reversible Markov processes, especialy in short times. The models studied are simple enough to get an explicit expression of the L2 distance betweeen the semigroup and the invariant measure throught time and to compare it with the corresponding reversible cases.

math.PR↗