Paris Dauphine University

About: Paris Dauphine University is a education organization based out in Paris, France. It is known for research contribution in the topics: Context (language use) & Population. The organization has 1766 authors who have published 6909 publications receiving 162747 citations. The organization is also known as: Paris Dauphine & Dauphine.

Averaging along irregular curves and regularisation of ODEs

Abstract: We consider the ordinary differential equation (ODE) $dx_{t} =b(t,x_{t} ) dt+ dw_{t}$ where $w$ is a continuous driving function and $b$ is a time-dependent vector field which possibly is only a distribution in the space variable. We quantify the regularising properties of an arbitrary continuous path $w$ on the existence and uniqueness of solutions to this equation. In this context we introduce the notion of $\rho$-\tmtextit{irregularity} and show that it plays a key role in some instances of the regularisation by noise phenomenon. In the particular case of a function $w$ sampled according to the law of the fractional Brownian motion of Hurst index $H \in (0,1)$, we prove that almost surely the ODE admits a solution for all $b$ in the Besov-H\~A{\P}lder space $B^{\alpha+1}_{\infty , \infty}$ with $\alpha >-1/2H$. If $\alpha >1-1/2H$ then the solution is unique among a natural set of continuous solutions. If $H>1/3$ and $\alpha >3/2-1/2H$ or if $\alpha >2-1/2H$ then the equation admits a unique Lipschitz flow. Note that when $\alpha <0$ the vector field $b$ is only a distribution, nonetheless there exists a natural notion of solution for which the above results apply.

Uniform Convergence to Equilibrium for Granular Media

Abstract: We study the long time asymptotics of a nonlinear, nonlocal equation used in the modelling of granular media We prove a uniform exponential convergence to equilibrium for degenerately convex and non convex interaction or confinement potentials, improving in particular results by J A Carrillo, R J McCann and C Villani The method is based on studying the dissipation of the Wasserstein distance between a solution and the steady state

Viability Kernels and Capture Basins of Sets Under Differential Inclusions

Abstract: This paper provides a characterization of viability kernels and capture basins of a target viable in a constrained subset as a unique closed subset between the target and the constrained subset satisfying tangential conditions or, by duality, normal conditions. It is based on a method devised by Helene Frankowska for characterizing the value function of an optimal control problem as generalized (contingent or viscosity) solutions to Hamilton--Jacobi equations. These abstract results, interesting by themselves, can be applied to epigraphs of functions or graphs of maps and happen to be very efficient for solving other problems, such as stopping time problems, dynamical games, boundary-value problems for systems of partial differential equations, and impulse and hybrid control systems, which are the topics of other companion papers.

Mobility Patterns

Abstract: We present a data model for tracking mobile objects and reporting the result of queries. The model relies on a discrete view of the spatio-temporal space, where the 2D space and the time axis are respectively partitioned in a finite set of user-defined areas and in constant-size intervals. We define a generic query language to retrieve objects that match mobility patterns describing a sequence of moves. We also identify a subset of restrictions to this language in order to express only deterministic queries for which we discuss evaluation techniques to maintain incrementally the result of queries. The model is conceptually simple, efficient, and constitutes a practical and effective solution to the problem of continuously tracking moving objects with sequence queries.