A positive radial product formula for the Dunkl kernel

It is an open conjecture that generalized Bessel functions associated with root systems have a positive product formula for nonnegative multiplicity parameters of the associated Dunkl operators. In this paper, a partial result towards this conjecture is proven, namely a positive radial product formula for the non-symmetric counterpart of the generalized Bessel function, the Dunkl kernel. Radial here means that one of the factors in the product formula is replaced by its mean over a sphere. The key to this product formula is a positivity result for the Dunkl-type spherical mean operator. It can also be interpreted in the sense that the Dunkl-type generalized translation of radial functions is positivity-preserving. As an application, we construct Dunkl-type homogeneous Markov processes associated with radial probability distributions.

     

Weighted Function Spaces and Dunkl Transform

We introduce first weighted function spaces on ${\mathbb{R}^d}$ using the Dunkl convolution that we call Besov-Dunkl spaces. We provide characterizations of these spaces by decomposition of functions. Next we obtain in the real line and in radial case on ${\mathbb{R}^d}$ weighted L ^p -estimates of the Dunkl transform of a function in terms of an integral modulus of continuity which gives a quantitative form of the Riemann-Lebesgue lemma. Finally, we show in both cases that the Dunkl transform of a function is in L ¹ when this function belongs to a suitable Besov-Dunkl space.     

The Dunkl intertwining operator

We consider Dunkl theory associated to a general Coxeter group G. A new characterization of the regularity of the weight k is given and a new construction, devoid of Kozul complex theory, of the Dunkl intertwining operator Vk is established. We apply our results to derive sharp estimates of the Dunkl kernel. We give explicit formula in the case of orthogonal positive root systems.     

Les équations de la chaleur et de Poisson pour le laplacien généralisé de Jacobi–Dunkl

We show that the heat equation for the Jacobi–Dunkl operator, has a solution in terms of a semigroup of Markovian operators with strictly positive kernel. This result is used to solve the Poisson equation and to introduce a new class of Markov processes on the real line. To cite this article: F. Chouchene et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

The approach to consistency in the analytic hierarchy process

In his first book on the Analytic Hierarchy Process, T. L. Saaty left open several mathematical questions about the structure of the set of positive reciprocal matrices. In this paper we consider three of these questions: Given an eigenvector and all matrices which give rise to it, can one go from one of them to any order by making small perturbations in the entries? Given two positive column vectors v and w is there a perturbation which carries the set of all positive reciprocal matrices with principal right eigenvector v to the set of positive reciprocal matrices with principal right eigenvector w? Does the set of positive reciprocal n×n matrices whose left and right principal eigenvectors are reciprocals coincide with the set of consistent matrices for n⩾4?     

Large deviation inequalities for supermartingales and applications to directed polymers in a random environment

We first show exponential large deviation inequalities for supermartingales, which are optimal in the independent and identically distributed (iid) case. We then apply them to establish new exponential concentration inequalities for the free energy of directed polymers in random environment; as consequences we obtain upper bounds of its rate of convergences (in probability, almost surely, and in $L^p$), and give an expression for the free energy in terms of free energies of some multiplicative cascades. To cite this article: Q. Liu, F. Watbled, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

The Divisible Sandpile at Critical Density

The divisible sandpile starts with i.i.d. random variables (“masses”) at the vertices of an infinite, vertex-transitive graph, and redistributes mass by a local toppling rule in an attempt to make all masses ≤  1. The process stabilizes almost surely if m < 1 and it almost surely does not stabilize if m > 1, where m is the mean mass per vertex. The main result of this paper is that in the critical case m = 1, if the initial masses have finite variance, then the process almost surely does not stabilize. To give quantitative estimates on a finite graph, we relate the number of topplings to a discrete bi-Laplacian Gaussian field.     

Howe duality in Dunkl superspace

In the framework of superspace in Clifford analysis for the Dunkl version, the Fischer decomposition is established for solutions of the Dunkl super Dirac operators. The result is general without restrictions on multiplicity functions or on super dimensions. The Fischer decomposition provides a module for the Howe dual pair G × osp(1|2) on the space of spinor valued polynomials with G the Coxeter group, while the generators of the Lie superspace reveal the naturality of the Fischer decomposition.     

The Heckman–Opdam Markov processes

We introduce and study the natural counterpart of the Dunkl Markov processes in a negatively curved setting. We give a semimartingale decomposition of the radial part, and some properties of the jumps. We prove also a law of large numbers, a central limit theorem, and the convergence of the normalized process to the Dunkl process. Eventually we describe the asymptotic behavior of the infinite loop as it was done by Anker, Bougerol and Jeulin in the symmetric spaces setting in (Iberoamericana 18: 41–97, 2002). Partially supported by the European Commission (IHP Network HARP 2002–2006).     

Monodromie du problème de Cauchy ramifiéMonodromy of the ramified Cauchy problem

In this Note, we study the monodromy of the ramified Cauchy problem for operators with multiple characteristics of constant multiplicity. More precisely, we give an estimation of the eigenvalues of the solution's monodromy, first with the assumptions of the theorem of Hamada–Leray–Wagschal, then with the assumptions of the theorem of Leichtnam. To cite this article: R. Camalès, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 639–642.     

Upper Bound for the Bethe–Sommerfeld Threshold and the Spectrum of the Poisson Random Hamiltonian in Two Dimensions

We consider the Schrödinger operator on ${\mathbb{R}^2}$ with a locally square-integrable periodic potential V and give an upper bound for the Bethe–Sommerfeld threshold (the minimal energy above which no spectral gaps occur) with respect to the square-integrable norm of V on a fundamental domain, provided that V is small. As an application, we prove the spectrum of the two-dimensional Schrödinger operator with the Poisson type random potential almost surely equals the positive real axis or the whole real axis, according as the negative part of the single-site potential equals zero or not. The latter result completes the missing part of the result by Ando et al. (Ann Henri Poincaré 7:145–160, 2006).     

Fourier-Jacobi type spherical functions for principal series representations of Sp(2, R)

In this paper, we study the Fourier-Jacobi type spherical functions on Sp(2, R) for irreducible principal series representations. We give the multiplicity theorem and an explicit formula for this function.     

Modularity in random regular graphs and lattices

Given a graph G, the modularity of a partition of the vertex set measures the extent to which edge density is higher within parts than between parts; and the modularity of G is the maximum modularity of a partition.We give an upper bound on the modularity of r-regular graphs as a function of the edge expansion (or isoperimetric number) under the restriction that each part in our partition has a sub-linear numbers of vertices. This leads to results for random r-regular graphs. In particular we show the modularity of a random cubic graph partitioned into sub-linear parts is almost surely in the interval (0.66, 0.88).The modularity of a complete rectangular section of the integer lattice in a fixed dimension was estimated in Guimer et. al. [R. Guimerà, M. Sales-Pardo and L.A. Amaral, Modularity from fluctuations in random graphs and complex networks, Phys. Rev. E 70 (2) (2004) 025101]. We extend this result to any subgraph of such a lattice, and indeed to more general graphs.     

Stochastic Nicholson’s blowflies delay differential equation with regime switching

In this paper, we investigate the global existence of almost surely positive solution to a stochastic Nicholson’s blowflies delay differential equation with regime switching, and give the estimation of the path. The results presented in this paper extend some corresponding results in Wang et al. Stochastic Nicholson’s blowflies delayed differential equations, Appl. Math. Lett. 87 (2019) 20–26 .     

Entropy, invertibility and variational calculus of adapted shifts on Wiener space

In this work we study the necessary and sufficient conditions for a positive random variable whose expectation under the Wiener measure is one, to be represented as the Radon-Nikodym derivative of the image of the Wiener measure under an adapted perturbation of identity with the help of the associated innovation process. We prove that the innovation conjecture holds if and only if the original process is almost surely invertible. We also give variational characterizations of the invertibility of the perturbations of identity and the representability of a positive random variable whose total mass is equal to unity. We prove in particular that an adapted perturbation of identity U=IW+u satisfying the Girsanov theorem, is invertible if and only if the kinetic energy of u is equal to the entropy of the measure induced with the action of U on the Wiener measure μ, in other words U is invertible iff

     

Estimation non paramétrique de la régression avec variable explicative dans un espace métrique

We study a nonparametric regression estimator when the explanatory variable takes its values in a semi-metric space. We establish some asymptotic results and give upper bounds of the p-mean and the almost sure estimation errors under general conditions. We end by an application to the discrimination in a semi-metric space and illustrate the results by the example of Wiener process as an explanatory variable. To cite this article: S. Dabo-Niang, N. Rhomari, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Some new examples of Markov processes which enjoy the time-inversion property

In this paper we give a sufficient condition on the semi group densities of an homogeneous Markov process taking values in ⁿ which ensures that it enjoys the time-inversion property. Our condition covers all previously known examples of Markov processes satisfying this property. As new examples we present a class of Markov processes with jumps, the Dunkl processes and their radial parts.Mathematics Subject Classification (2000): 60J25, 60J60, 60J65, 60J99     

Propriété de Liouville et équation de Poisson pour le laplacien généralisé de Dunkl

Let Δ_k be the Dunkl generalized Laplacian associated to a root system R of $\text{R}{}^{\mathrm{N}}$ and a non-negative function k defined on R and invariant by the Weyl group. In this Note, we show that this differential-difference operator on $\text{R}{}^{\mathrm{N}}$ satisfies the Liouville property, then we solve the Poisson equation Δ_ku=?f by using a generalized Fourier analysis method. To cite this article: L. Gallardo, L. Godefroy, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

On convergence of solutions to difference equations with additive perturbations

Various types of stabilizing controls lead to a deterministic difference equation with the following property: once the initial value is positive, the solution tends to the unique positive equilibrium. Introducing additive perturbations can change this picture: we give examples of difference equations experiencing additive perturbations which have solutions staying around zero rather than tending to the unique positive equilibrium. When perturbations are stochastic with a bounded support, we give an upper estimate for the probability that the solution can stay around zero. Applying extra conditions on the behaviour of the map function f at zero or on the amplitudes of stochastic perturbations, we prove that the solution tends to the unique positive equilibrium almost surely. In particular, this holds either for all amplitudes when the right derivative of the map f at zero exceeds one or, independently of the behaviour of f at zero, when the amplitudes are not square summable.     

Positive solutions of m-point boundary value problems for a class of nonlinear fractional differential equations

This paper deals with the existence and multiplicity of positive solutions for a class of nonlinear fractional differential equations with m-point boundary value problems. We obtain some existence results of positive solution by using the properties of the Green’s function, u ₀-bounded function and the fixed point index theory under some conditions concerning the first eigenvalue with respect to the relevant linear operator.