arXiv subjects

Florent Malrieu

Publications and source records attributed to Florent Malrieu.

21 records · Page 2Linked to original sources

arXiv2009

Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation

We consider a Vlasov-Fokker-Planck equation governing the evolution of the density of interacting and diffusive matter in the space of positions and velocities. We use a probabilistic interpretation to obtain convergence towards equilibrium in Wasserstein distance with an explicit exponential rate. We also prove a propagation of chaos property for an associated particle system, and give rates on the approximation of the solution by the particle system. Finally, a transportation inequality for the distribution of the particle system leads to quantitative deviation bounds on the approximation of the equilibrium solution of the equation by an empirical mean of the particles at given time.

math.PR↗

arXiv2008

Propagation of chaos and Poincaré inequalities for a system of particles interacting through their cdf

In the particular case of a concave flux function, we are interested in the long time behaviour of the nonlinear process associated to the one-dimensional viscous scalar conservation law. We also consider the particle system obtained by remplacing the cumulative distribution function in the drift coefficient of this nonlinear process by the empirical cdf. We first obtain trajectorial propagation of chaos result. Then, Poincaré inequalities are used to get explicit estimates concerning the long time behaviour of both the nonlinear process and the particle system.

math.PR↗

arXiv2006

Logarithmic Sobolev Inequalities for Inhomogeneous Markov Semigroups

We investigate the dissipativity properties of a class of scalar second order parabolic partial differential equations with time-dependent coefficients. We provide explicit condition on the drift term which ensure that the relative entropy of one particular orbit with respect to some other one decreases to zero. The decay rate is obtained explicitly by the use of a Sobolev logarithmic inequality for the associated semigroup, which is derived by an adaptation of Bakry's $Γ-$ calculus. As a byproduct, the systematic method for constructing entropies which we propose here also yields the well-known intermediate asymptotics for the heat equation in a very quick way, and without having to rescale the original equation.

math.PR↗