arXiv subjects

Jean-François Jabir

Publications and source records attributed to Jean-François Jabir.

4 recordsLinked to original sources

arXiv2026

Kinetic SDEs and kinetic Fokker-Planck equations with critical Morrey-type drifts, and local blow-ups

We develop a Sobolev theory for the kinetic Fokker-Planck equation subject to a singular Morrey-form $L^\infty+M^q_K$ force field which can produce local blow-ups due to strong attraction to a submanifold. Our approach is based on the fractional representation for the Duhamel series of the equation and on establishing kinetic Adams-type estimates. Accordingly, we construct weak solutions of Brownian-driven kinetic SDEs in a critical Morrey set-up, obtain kinetic Krylov-type estimates and a strongly continuous Feller propagator, and establish uniqueness criteria for a large class of weak solutions.

math.PR↗

arXiv2026

Freidlin-Wentzell collision-laws between self-stabilizing diffusions

The present work investigates the asymptotic behaviours at the zero-noise limit of the first near collision-time and first near collision-location between a pair of independent $d$-dimensional Brownian-driven self-stabilizing (McKean-Vlasov type) diffusions. These asymptotic are considered in a peculiar setting where the systems evolve in a bi-stable landscape and collisions are only triggered by the combined action of the Brownian noises. As the Brownian perturbations fade away, we show that the near collision-time increases at an explicit exponential rate and that related collision-locations persist in specific regions of the space. These results are mainly derived by tailoring classical Freidlin-Wentzell's exit-time (and exit-location) estimates into collision estimates. Similar asymptotic are established for related mean-field interacting particle system approximation, and for the one-dimensional case (where true collisions can be examined

math.PR↗

arXiv2021

Mean-Field Neural ODEs via Relaxed Optimal Control

We develop a framework for the analysis of deep neural networks and neural ODE models that are trained with stochastic gradient algorithms. We do that by identifying the connections between control theory, deep learning and theory of statistical sampling. We derive Pontryagin's optimality principle and study the corresponding gradient flow in the form of Mean-Field Langevin dynamics (MFLD) for solving relaxed data-driven control problems. Subsequently, we study uniform-in-time propagation of chaos of time-discretised MFLD. We derive explicit convergence rate in terms of the learning rate, the number of particles/model parameters and the number of iterations of the gradient algorithm. In addition, we study the error arising when using a finite training data set and thus provide quantitive bounds on the generalisation error. Crucially, the obtained rates are dimension-independent. This is possible by exploiting the regularity of the model with respect to the measure over the parameter space.

math.PR↗

arXiv2013

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.

math-ph↗