## P. LAURENÇOT, S. MISCHLER Global existence for the discrete diffusive coagulation-fragmentation equations in $L^1$

Reference: DMA-01-01 (January 2001)

Source:dvi, ps

Abstract: Existence of global weak solutions to the discrete coagulation-fragmentation equations with diffusion is proved under general assumptions on the coagulation and fragmentation coefficients. Unlike previous works requiring $L^\infty$-estimates, an $L^1$-approach is developed here which relies on weak and strong compactness methods in $L^1$.

AMS Classification :****
Keywords :****

## B. ENRIQUEZ On some universal algebras associated to the category of Lie bialgebras

Reference: DMA-01-02 (February 2001)

Source:dvi, ps

Abstract: In our previous work (math/0008128), we studied the set $\Quant(\KK)$ of all universal quantization functors of Lie bialgebras over a field $\KK$ of characteristic zero, compatible with the operations of taking duals and doubles. We showed that $\Quant(\KK)$ is canonically isomorphic to a product $\cG_0(\KK) \times \Sha(\KK)$, where $\cG_0(\KK)$ is a universal group and $\Sha(\KK)$ is a quotient set of a set $\cB(\KK)$ of families of Lie polynomials by the action of a group $\cG(\KK)$. We prove here that $\cG_0(\KK)$ is equal to the multiplicative group $1 + \hbar \KK[[\hbar]]$. So $\Quant(\KK)$ is as close as it can be' to $\Sha(\KK)$. We also prove that the only universal derivations of Lie bialgebras are multiples of the composition of the bracket with the cobracket. Finally, we prove that the stabilizer of any element of $\cB(\KK)$ is reduced to the $1$-parameter subgroup generated by the corresponding square of the antipode'.

AMS Classification :****
Keywords :****

## D. HARARI Groupes alg\'ebriques et points rationnels

Reference: DMA-01-03 (March 2001)

Source:dvi, ps

Abstract: Soit $X$ une vari\'et\'e alg\'ebrique d\'efinie sur un corps de nombres. Nous comparons diverses obstructions cohomologiques (ab\'eliennes et non-ab\'eliennes) au principe de Hasse et \a l'approximation faible sur $X$ \a l'obstruction de Manin.

AMS Classification :****
Keywords :****

## S. HERSONSKY, F. PAULIN Diophantine Approximation in Negatively Curved Manifolds and in the Heisenberg Group (with an appendix: Diophantine Approximation on Hyperbolic Surfaces, by J. Parkkonen and F. Paulin)

Reference: DMA-01-04 (February 2001)

Source:ps

Abstract: This paper is a survey of the work of the authors [20,1,21], with a new application to Diophantine approximation in the Heisenberg group. The Heisenberg group, endowed with its Carnot Caratheodory metric, can be seen as the space at infinity of the complex hyperbolic space (minus one point). The rational approximation on the Heisenberg group can be interpreted and developed using arithmetic subgroups of $SU(n,1)$. In the appendix, the case of hyperbolic surfaces is developed by Jouni Parkkonen and the second author.

AMS Classification :****
Keywords :****

## A.-M. AUBERT, C. CUNNINGHAM Adic algebraic groups, parahoric subgroups and fundamental strata

Reference: DMA-01-05 (March 2001)

Source:dvi, ps

Abstract: This paper establishes basic properties of adic algebraic groups, adic parahoric groups and fundamental strata. The latter recovers the notion of K-types for $p$-adic groups as certain $l$-adic sheaf complexes on an adic parahoric group.

AMS Classification :****
Keywords :****

## P. LAURENÇOT, S. MISCHLER The continuous coagulation-fragmentation equations with diffusion

Reference: DMA-01-06 (July 2001)

Source:ps

Abstract: Existence of global weak solutions to the continuous coagulation-fragmentation equations with diffusion is investigated when the kinetic coefficients satisfy a detailed balance condition or the coagulation coefficient enjoys a monotonicity condition. Our approach relies on weak and strong compactness methods in $L^1$ in the spirit of the DiPerna-Lions theory for the Boltzmann equation. Under the detailed balance condition the large time behaviour is also studied.

AMS Classification :****
Keywords :****

## P. MARCHAL Loop-erased random walks and heaps of cycles

Reference: DMA-01-07 (March 2001)

Source:dvi, ps

Abstract: We use the combinatorial structure of {\it heap of cycles}~\cite{v} to study three algorithms related to loop-erased random walks. This includes Wilson's algorithm~\cite{pw}, which constructs random spanning trees, a related algorithm generating random Hamiltonian cycles~\cite{m1} and a third algorithm yielding a random, non-biased occupation measure in time comparable with the cover time.

AMS Classification :60J10, 60C05
Keywords :loop-erased random walk, spanning tree, occupation measure, cover time

## P. MARCHAL Self-repelling processes on the boundary of a regular tree

Reference: DMA-01-08 (March 2001)

Source:dvi, ps

Abstract: We construct the path of a self-repelling version of stable'' Lévy processes on \Z. This construction uses the tree representation of the paths of these processes, which was introduced in~\cite{m} for the study of multiple points. We compute the Hausdorff dimension of the range and the critical parameter for the self-avoiding property.

AMS Classification :****
Keywords :****

## J. BELLAICHE Relèvement des formes modulaires de Picard

Reference: DMA-01-09 (March 2001)

Source:ps

Abstract: Dans cet article, nous développons une version arithm\'etique du développement de Fourier-Jacobi des formes modulaires de Picard, et nous prouvons que toute forme modulaire de Picard en caractéristique $p$ provient par réduction d'une forme modulaire de Picard en caractéristique nulle. Nous utilisons ces résultats pour construire des formes modulaires en caractéristique nulle dont le développement de Fourier-Jacobi est congrue à 1 modulo $p$. (Ce preprint est une version l\'eg\`erement remani\'ee du preprint dma-00-17).\\ In this paper, we set up the arithmetic theory of Fourier-Jacobi extension for Picard modular forms, and we prove that every Picard modular form in characterstic $p$ come by reduction modulo $p$ from a Picard modular form in characteristic 0. We use these results to construct some modular forms in characteristic 0 with Fourier-Jacobi developpement is equal to 1 modulo $p$. (This preprint is a slightly changed version of the preprint dma-00-17)

AMS Classification :****
Keywords :****

## Y. BARAUD Model selection for regression on a random design

Reference: DMA-01-10 (April 2001)

Source:dvi, ps

Abstract: We consider the problem of estimating an unknown regression function when the design is random with values in $\R^k$. For this purpose, we consider the least-squares estimator built on a linear space which has been selected from the data among a collection of those given in advance. This data driven selection procedure is performed via the minimization of a penalized model selection criterion. We provide non asymptotic risk bounds for this penalized least-squares estimator (PLSE) from which we deduce adaptivity properties. When the regression function is additive we provide an additive estimator which converges at the same rate as it does when $k=1$.

AMS Classification :Primary 62G07; Secondary 62J02
Keywords :Nonparametric regression, Least-squares estimators, Penalized least-squares estimators, Model selection, Adaptive estimation, Additive functions

## P.-E. JABIN, F. OTTO, B. PERTHAME Line--energy Ginzburg--Landau models: zero--energy states

Reference: DMA-01-11 (April 2001)

Source:dvi, ps

Abstract: We consider a class of two--dimensional Ginzburg--Landau problems which are characterized by energy density concentrations on a one--dimensional set. In this paper, we investigate the states of vanishing energy. We classify these zero--energy states in the whole space: They are either constant or a vortex. A bounded domain can sustain a zero--energy state only if the domain is a disk and the state a vortex. Our proof is based on specific entropies which lead to a kinetic formulation, and on a careful analysis of the corresponding weak solutions by the method of characteristics.

AMS Classification :35B65, 35J60, 35L65, 74G65, 82D30.
Keywords :Ginzburg--Landau energy, vortices, kinetic formulation, ferromagnetism.

## F. GOLSE, C.D. LEVERMORE Stokes-Fourier and Acoustic Limits for the Boltzmann Equation: Convergence Proofs

Reference: DMA-01-12 (May 2001)

Source:dvi, ps

Abstract: We establish a Stokes-Fourier limit for the Boltzmann equation considered over any periodic spatial domain of dimension two or more. Appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations that globally in time converge weakly to a unique limit governed by a solution of Stokes-Fourier motion and heat equations provided that the fluid moments of their initial fluctuations converge to appropriate $L^2$ initial data of the Stokes-Fourier equations. Both the motion and heat equations are both recovered in the limit by controlling the fluxes and the local conservation defects of the DiPerna-Lions solutions with dissipation rate estimates. The scaling of the fluctuations with respect to Knudsen number is essentially optimal. The assumptions on the collision kernel are little more than those required for the DiPerna-Lions theory and that the viscosity and heat conduction are finite. For the acoustic limit, these techniques also remove restrictions to bounded collision kernels and improve the scaling of the fluctuations. Both weak limits become strong when the initial fluctuations converge entropically to appropriate $L^2$ initial data.

AMS Classification :****
Keywords :****

## B. PERTHAME, C. SIMEONI A kinetic scheme for the Saint-Venant system with a source term

Reference: DMA-01-13 (May 2001)

Source:ps

Abstract: The aim of this paper is to present a numerical scheme to compute Saint-Venant equations with a source term, due to the bottom topography, in a one-dimensional framework, which satisfies some theoretical properties: it preserves the steady state of still water, satisfies an entropy inequality, preserves the non-negativity of the height of water and remains stable with a discontinous bottom. This is achieved by means of a kinetic approach to the system, which is the departing point of the method developed here. In this context, we use a natural description of the microscopic behaviour of the system to define numerical fluxes at the interfaces of an unstructured mesh. We also use the concept of cell centered conservative quantities (as usual in finite volume method) and upwind interfacial sources as advocated by several authors. We show, analytically and also by means of some numerical results, that the above properties are satisfied.

AMS Classification :
Keywords :Saint-Venant system, finite volume method, upwind interfacial sources, kinetic schemes

## R. de la BRETECHE Problèmes extrémaux concernant les bases additives

Reference: DMA-01-14 (May 2001)

Source:dvi, ps

Abstract: Soit ${\cal A}$ une base additive exact d'ordre $h$. Nous \'etablissons une nouvelle majoration en fonction de $h$ de l'ordre exact de la base ${\cal A}\setminus \{a\}$ pour tout \'el\'ement $a$ de ${\cal A}$ sauf pour au plus un nombre fini. Ceci permet de montrer la conjecture de Grekos pour tout $h\geq 2$.\\ Let ${\cal A}$ be an exact additive basis of order $h$. We state a new bound of the exact order of the basis ${\cal A}\setminus \{a\}$ for all $a$ of ${\cal A}$ except for a finite number. This proves the Grekos' conjecture for all $h\geq 2$.

AMS Classification :11B13

## J. DENEF, F. LOESER Macdonald integrals and monodromy

Reference: DMA-01-15 (May 2001)

Source:ps

Abstract: We prove several results on monodromies associated to Macdonald integrals, that were used in our previous work on the finite field analogue of a conjecture of Macdonald. We also give a new proof of our formula expressing recursively the zeta function of the local monodromy at the origin of the discriminant of a finite Coxeter group in terms of the degrees of the group.

AMS Classification :****
Keywords :****

## P. LAURENÇOT, S. MISCHLER From the Becker-Döring to the Lifshitz-Slyozov-Wagner equations

Reference: DMA-01-16 (July 2001)

Source:ps

Abstract: Connections between two classical models of phase transitions, the Becker-D\"oring (BD) equations and the Lifshitz-Slyozov-Wagner (LSW) equation, are investigated. Homogeneous coefficients are first considered and a previous works by Penrose and Collet, Goudon, Poupaud \& Vasseur. Convergence of the solutions to these rescaled BD equations towards a solution to the LSW equation is shown. For general coefficients an approach in the spirit of numerical analysis allows to approximate the Lifshitz-Slyozov equation by a sequence of BD equations. A new uniqueness result for the BD equations is also provided.

AMS Classification :****
Keywords :****

## P.-E. JABIN Various levels of models for aerosols

Reference: DMA-01-17 (September 2001)

Source:dvi, ps

Abstract: Two limit behaviours of a simple model of aerosol are considered. The only force acting on aerosol particles is a friction due to the flow of gas. It is first proved that in the limit of an infinite friction coefficient, the particles are simply advected by the gas. Then we consider very dilute sprays of aerosol, {\it i.e.} with distribution functions which are monokinetic (Dirac mass in velocity). This approach leads to a macroscopic system with a free-boundary problem.

AMS Classification :****
Keywords :****

## P. Laurençot, S. Mischler From the discrete to the continuous coagulation-fragmentation equations

Reference: DMA-01-18 (September 2001)

Source:dvi, ps

Abstract: The connection between the discrete and the continuous coagulation-frag\-men\-ta\-tion models is investigated. A weak stability principle relying on \textit{a priori} estimates and weak compactness in $L^1$ is developed for the continuous model. We approximate the continuous model by a sequence of discrete models and, writing the discrete models as modified continuous ones, we prove the convergence of the latter towards the former with the help of the above mentioned stability principle. Another application of this stability principle is the convergence of an explicit time and size discretisation of the continuous coagulation-fragmentation model.

AMS Classification :****
Keywords :****

## M. Escobedo, S. Mischler, B. Perthame Gelation in coagulation and fragmentation models

Reference: DMA-01-19 (September 2001)

Source:dvi, ps

Abstract: Rates of decay for the total mass of the solutions to Smoluchovski's equation with homogeneous kernels of degree $\lambda >1$ are proved. That implies that gelation always occurs. Morrey estimates from below and from above on solutions around the gelation time are also obtained which are in agreement with previously known formal results on the profile of solutions at gelling time. The same techniques are applied to the coagulation-fragmentation model for which gelation is established in some particular cases.

AMS Classification :****
Keywords :****

## M. Escobedo, S. Mischler, J.~J.~L. Velazquez Asymptotic description of dirac mass formation in kinetic equations for quantum particles

Reference: DMA-01-20 (September 2001)

Source:dvi, ps

Abstract: In this paper we describe in a detailed manner the way in which a dirac mass is generated asymptotically in time for solutions of a kinetic equation for quantum particles.

AMS Classification :****
Keywords :****

## S. BARANNIKOV Semi-infinite variations of Hodge structures and integrable hierarchies of KdV type

Reference: DMA-01-21 (September 2001)

Source:dvi, ps

Abstract: We consider integrable KdV type hierarchy associated naturally with arbitrary semi-simple Frobenius manifold. Equations defining hierarchy have a Lax form and admit a bihamiltonian description. This hierarchy is responsable for extension of corresponding semi-infinite variation of Hodge structures by generating variations along higher times

AMS Classification :****
Keywords :****

## H. ZAAG One dimensional behavior of singular $N$ dimensional solutions of semilinear heat equations

Reference: DMA-01-22 (September 2001)

Source:dvi, ps

Abstract: We consider $u(x,t)$ a blow-up solution of $u_t= \Delta u +|u|^{p-1}u$ where $u:\R^N\times[0, T)\to \R$, $p>1$, $(N-2)p3$, then $u$ is {\it very} close to a superposition of one dimensional solutions as functions of the distance to $S$.

AMS Classification :****
Keywords :****

## T. Duquesne, J.-F. Le Gall Random Trees, L\'evy Processes and Spatial Branching Processes

Reference: DMA-01-23 (November 2001)

Source:dvi, ps

Abstract: ****

AMS Classification :****
Keywords :****

## S. Hersonsky, F. Paulin Counting orbit points in coverings of negatively curved manifolds and Hausdorff dimension of cusp excursions

Reference: DMA-01-24 (October 2001)

Source:dvi, ps

Abstract: We study the growth of fibers of coverings of pinched negatively curved Riemannian manifold. The applications include counting estimates for horoballs in the universal cover of geometrically finite manifolds with cusps. Continuing our work on diophantine approximation in negatively curved manifolds started in \cite{HP}, we prove a Khintchine-Sullivan type theorem giving the Hausdorff measure of the geodesic lines starting from a cusp that are well approximated by the cusp returning ones.

AMS Classification :53 C 22, 11 J 06, 30 F 40, 11 J 70.
Keywords :rational geodesic, negative curvature, cusp, horoball

## P. BIANE Characters of symmetric groups and free cumulants

Reference: DMA-01-25 (November 2001)

Source:dvi, ps

Abstract: We investigate Kerov's formula expressing the normalized irreducible characters of symmetric groups evaluated on a cycle, in terms of the free cumulants of the associated Young diagrams.

AMS Classification :****
Keywords :****

## P. BIANE Parking functions of types A and B

Reference: DMA-01-26 (November 2001)

Source:dvi, ps

Abstract: The lattice of noncrossing partitions can be embedded into the Cayley graph of the symmetric group. This allows us to rederive connections between noncrossing partitions and parking functions. We use an analogous embedding for type B non-crossing partitions in order to answer a question raised by R. Stanley on the edge labeling of the type B non-crossing partitions lattice.

AMS Classification :****
Keywords :****

## M. ESCOBEDO, S. MISCHLER, M.A. VALLE Homogeneous Boltzmann equation for quantum and relativistic particles

Reference: DMA-01-27 (November 2001)

Source:dvi, ps

Abstract: ****

AMS Classification :****
Keywords :****

## Y. BARAUD Confidence balls in nonparametric Gaussian regression

Reference: DMA-01-28 (November 2001)

Source:ps

Abstract: Starting from the observation of an $\R^n$-Gaussian vector of mean $f$ and covariance matrix $\var I$ ($I$ is the identity matrix), we propose two methods for building an Euclidean confidence ball around $f$, with prescribed probability of coverage. The first method is based on prior information on the variance $\var$ and is free from any assumption on $f$. In contrast, the other method is based on prior information on the ratio signal-noise, $f/\sigma$. For each $n$, we describe the nonasymptotic properties of the so-defined confidence balls and show their optimality with respect to some criteria.

AMS Classification :****
Keywords :Confidence ball, nonparametric regression, hypothesis testing, estimation

## C. SABOT Spectral properties of hierarchical lattices and iteration of rational maps

Reference: DMA-01-29 (November 2001)

Source:ps

Abstract: In this text we are interested in spectral properties of discrete Laplace operators defined on lattices based on finitely-ramified self-similar sets. The basic example is the lattice based on the Sierpinski gasket. We introduce a new renormalization map which appears to be a rational self-map of a compact complex manifolds. We relate some characteristics of its dynamics with some characteristics of the spectrum of our operator. More specifically, we give an explicite formula for the density of states in terms of the Green current of the map, and we relate the indeterminacy points of the map with the so-called Neuman-Dirichlet eigenvalues which lead to eigenfunctions with compact support on the unbounded lattice. Depending on the asymptotic degree of the map we can prove drastic different spectral properties of the operator. Hence, this work aims at a generalization and a better understanding of the initial work of physisits Rammal and Toulouse on the Sierpinski gasket.

AMS Classification :82B44(32H50,28A80)
Keywords :Spectral theory of Schr\"odinger operators, pluricomplex dynamics, analysis on self-similar sets.

## C. SABOT Schr\"odinger operators on fractal lattices with random blow-ups

Reference: DMA-01-30 (November 2001)

Source:ps

Abstract: Starting from a finitely ramified self-similar set $X$ we can construct an unbounded set $X_\infi$ by blowing-up the initial set $X$. We consider random blow-ups and prove elementary properties of the spectrum of the natural Laplace operator on $X_\infi$ (and on the associated lattice). We prove that the spectral type of the operator is almost surely deterministic with the blow-up and that the spectrum coincides with the support of the density of states almost surely (actually our result is more precise). We also prove that if the density of states is completely created by the so-called Neuman-Dirichlet eigenvalues, then almost surely the spectrum is pure point with compactly supported eigenfunctions.

AMS Classification :82B44(60H25,28A80)
Keywords :Spectral theory of Schr\"odinger operators, random self-adjoint operators, analysis on self-similar sets.

## I. MARIN Quotients infinit\'esimaux du groupe de tresses

Reference: DMA-01-31 (November 2001)

Source:ps

Abstract: We define and study infinitesimal analogues of the main quotients of the group algebra of the Artin's groups, namely the Temperly-Lieb, Hecke and Birman-Wenzl-Murakami algebras, in terms of KZ-systems. These analogues are Hopf algebras which correspond to reductive groups : we give then a general framework for the study of representations deduced from the classical representations of $B_n$ through tensor constructions. We use this to analyse representations related to the Burau representation, and we fully decompose the infinitesimal Temperly-Lieb algebra. As a by-product, we obtain several irreducibility properties.

AMS Classification :****
Keywords :****

## Y. BARAUD, S. HUET, B. LAURENT Tests for convex hypotheses

Reference: DMA-01-32 (November 2001)

Source:ps

Abstract: In this paper we consider the problem of testing that the mean of a Gaussian vector in $\R^n$ belongs to a convex set. The testing procedure we propose is based on multiple testing. We show that the test achieves its nominal level, and for each $n$ we describe a class of vectors on which the test is powerful. We apply the procedure to test that the regression function in a Gaussian regression model is positive, increasing, convex or more generally satisfies a differential inequality. The tests do not rely on any prior information on the regression function. Separation rates over classes of smooth functions are established and a simulation study evaluates some of the procedures for testing monotonicity.

AMS Classification :Primary 62G10; Secondary 62G20.
Keywords :tests of qualitative hypotheses, nonparametric test, test of positivity, test of monotonicity, test of convexity, rate of testing, Gaussian regression

## G-Q. CHEN, B. PERTHAME Well-Posedness for Non-Isotropic Degenerate Parabolic-Hyperbolic Equations

Reference: DMA-01-33 (December 2001)

Source:ps

Abstract: We develop a well-posedness theory for solutions in $L^1$ to the Cauchy problem of general degenerate parabolic-hyperbolic equations with non-isotropic nonlinearity. A new notion of entropic and kinetic solutions, and a corresponding kinetic formulation is developed which extends the hyperbolic case. The notion of kinetic solutions applies to more general situations than that of entropy solutions; and its advantage is that the kinetic equations in the kinetic formulation are well defined even when the macroscopic fluxes are not locally integrable, so that $L^1$ is a natural space on which the kinetic solutions are posed. Based on this notion, we develop a new, simpler, more effective approach to prove the contraction property of kinetic solutions in $L^1$, especially including entropy solutions. It includes a new ingredient, a chain rule type condition, which makes it different from the isotropic case.

AMS Classification :35K65, 35K10, 35B30, 35D05
Keywords :Kinetic solutions, entropy solutions, kinetic formulation, degenerate parabolic equations, convection-diffusion, non-isotropic diffusion, stability, existence, well-posedness

Commentaires à preprints@dma.ens.fr.