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.

Physical Brownian motion in a magnetic field as a rough path

Abstract: The indefinite integral of the homogenized Ornstein-Uhlenbeck process is a well-known model for physical Brownian motion, modelling the behaviour of an object subject to random impulses [L. S. Ornstein, G. E. Uhlenbeck: On the theory of Brownian Motion. In: Physical Review. 36, 1930, 823-841]. One can scale these models by changing the mass of the particle and in the small mass limit one has almost sure uniform convergence in distribution to the standard idealized model of mathematical Brownian motion. This provides one well known way of realising the Wiener process. However, this result is less robust than it would appear and important generic functionals of the trajectories of the physical Brownian motion do not necessarily converge to the same functionals of Brownian motion when one takes the small mass limit. In presence of a magnetic field the area process associated to the physical process converges - but not to L\\'evy's stochastic area. As this area is felt generically in settings where the particle interacts through force fields in a nonlinear way, the remark is physically significant and indicates that classical Brownian motion, with its usual stochastic calculus, is not an appropriate model for the limiting behaviour. We compute explicitly the area correction term and establish convergence, in the small mass limit, of the physical Brownian motion in the rough path sense. The small mass limit for the motion of a charged particle in the presence of a magnetic field is, in distribution, an easily calculable, but \"non-canonical\" rough path lift of Brownian motion. Viewing the trajectory of a charged Brownian particle with small mass as a rough path is informative and allows one to retain information that would be lost if one only considered it as a classical trajectory. We comment on the importance of this point of view.

Partial differential equations and image processing

Abstract: We review the PDE methods for image processing developed at the Ceremade, the Universitat de Illes Balears, and the Universidad de Las Palmas de Gran Canaria. We first recall the axiomatic approach of the theory of multiscale analysis of images and try at the same times to see how it can be extended to the problem of color image processing. After stating the theorems that follow from this theory, we discuss some examples of PDEs and show results. >

Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations

Abstract: We present exponential error estimates and demonstrate an algebraic convergence rate for the homogenization of level-set convex Hamilton-Jacobi equations in i.i.d. random environments, the first quantitative homogenization results for these equations in the stochastic setting. By taking advantage of a connection between the metric approach to homogenization and the theory of first-passage percolation, we obtain estimates on the fluctuations of the solutions to the approximate cell problem in the ballistic regime (away from flat spot of the effective Hamiltonian). In the sub-ballistic regime (on the flat spot), we show that the fluctuations are governed by an entirely different mechanism and the homogenization may proceed, without further assumptions, at an arbitrarily slow rate. We identify a necessary and sufficient condition on the law of the Hamiltonian for an algebraic rate of convergence to hold in the sub-ballistic regime and show, under this hypothesis, that the two rates may be merged to yield comprehensive error estimates and an algebraic rate of convergence for homogenization. Our methods are novel and quite different from the techniques employed in the periodic setting, although we benefit from previous works in both first-passage percolation and homogenization. The link between the rate of homogenization and the flat spot of the effective Hamiltonian, which is related to the nonexistence of correctors, is a purely random phenomenon observed here for the first time.