Joaquin Fontbona

Quantitative estimates for the long-time behavior of an ergodic variant of the telegraph process

Motivated by stability questions on piecewise-deterministic Markov models of bacterial chemotaxis, we study the long-time behavior of a variant of the classic telegraph process having a nonconstant jump rate that induces a drift towards the origin. We compute its invariant law and show exponential ergodicity, obtaining a quantitative control of the total variation distance to equilibrium at each instant of time. These results rely on an exact description of the excursions of the process away from the origin and on the explicit construction of an original coalescent coupling for both the velocity and position. Sharpness of the obtained convergence rate is discussed.

Quantitative propagation of chaos for generalized Kac particle systems.

Abstract We study a class of one dimensional particle systems with binary interactions of Birdtype, which includes Kac’s simplified model of the Boltzmann equation and some kineticmodelsfortheevolutionofwealthdistribution. Weobtainexplicitratesofconvergence,asthetotalnumberofparticlesgoesto∞,fortheWassersteindistancebetweenthelawofaparticleanditslimitinglaw,whichdependlinearlyontime. Theproofisbasedonanovelcoupling between the particle system and a suitable system of non-independent nonlinearprocesses,aswellasonrecentsharpestimatesforempiricalmeasures. Keywords: propagation of chaos, jump processes, Kac equation, wealth distribution, Birdparticlesystem,Wassersteindistance,optimalcoupling. AMSclassification(2010): 60K35,82C22,82C40. 1 Introduction and main result Thekineticequation Weconsiderthecollection( P t ) t ≥0 ofprobabilitymeasuresonR,solutionofthefollowingnonlinearkinetic-typeequation: ∂ t P t = − P t + Q + ( P t ) . (1)HereQ + isageneralizedWildconvolution,whichassociateswitheverymeasure

Non local Lotka-Volterra system with cross-diffusion in an heterogeneous medium

We introduce a stochastic individual model for the spatial behavior of an animal population of dispersive and competitive species, considering various kinds of biological effects, such as heterogeneity of environmental conditions, mutual attractive or repulsive interactions between individuals or competition between them for resources. As a consequence of the study of the large population limit, global existence of a nonnegative weak solution to a multidimensional parabolic strongly coupled model of competing species is proved. The main new feature of the corresponding integro-differential equation is the nonlocal nonlinearity appearing in the diffusion terms, which may depend on the spatial densities of all population types. Moreover, the diffusion matrix is generally not strictly positive definite and the cross-diffusion effect allows for influences growing linearly with the subpopulations’ sizes. We prove uniqueness of the finite measure-valued solution and give conditions under which the solution takes values in a functional space. We then make the competition kernels converge to a Dirac measure and obtain the existence of a solution to a locally competitive version of the previous equation. The techniques are essentially based on the underlying stochastic flow related to the dispersive part of the dynamics, and the use of suitable dual distances in the space of finite measures.

Local Existence of Analytical Solutions to an Incompressible Lagrangian Stochastic Model in a Periodic Domain

We consider an incompressible kinetic Fokker Planck equation in the flat torus, which is a simplified version of the Lagrangian stochastic models for turbulent flows introduced by S.B. Pope in the context of computational fluid dynamics. The main difficulties in its treatment arise from a pressure type force that couples the Fokker Planck equation with a Poisson equation which strongly depends on the second order moments of the fluid velocity. In this paper we prove short time existence of analytic solutions in the one-dimensional case, for which we are able to use techniques and functional norms that have been recently introduced in the study of a related singular model.

Ray–Knight representation of flows of branching processes with competition by pruning of Lévy trees

We introduce flows of branching processes with competition, which describe the evolution of general continuous state branching populations in which interactions between individuals give rise to a negative density dependence term. This generalizes the logistic branching processes studied by Lambert (Ann Appl Probab 15(2):1506–1535, 2005). Following the approach developed by Dawson and Li (Ann Probab 40(2):813–857, 2012), we first construct such processes as the solutions of certain flow of stochastic differential equations. We then propose a novel genealogical description for branching processes with competition based on interactive pruning of Lévy-trees, and establish a Ray–Knight representation result for these processes in terms of the local times of suitably pruned forests.

Quantitative Uniform Propagation of Chaos for Maxwell Molecules

We prove propagation of chaos at explicit polynomial rates in Wasserstein distance

A random space-time birth particle method for 2d vortex equations with external field

{\mathcal{W}_{2}}

Measurability of optimal transportation and convergence rate for Landau type interacting particle systems

W2 for Kac’s N-particle system associated with the spatially homogeneous Boltzmann equation for Maxwell molecules. Our approach is mainly based on novel probabilistic coupling techniques. Combining them with recent stabilization results for the particle system we obtain, under suitable moments assumptions on the initial distribution, a uniform-in-time estimate of order almost