Convergence In L1 Research Articles

Weak and strong error analysis for mean-field rank-based particle approximations of one-dimensional viscous scalar conservation laws

In this paper, we analyse the rate of convergence of a system of N interacting particles with mean-field rank-based interaction in the drift coefficient and constant diffusion coefficient. We first adapt arguments by Kolli and Shkolnikov (Ann. Probab. 46 (2018) 1042–1069) to check trajectorial propagation of chaos with optimal rate N−1/2 to the associated stochastic differential equations nonlinear in the sense of McKean. We next relax the assumptions needed by Bossy (Math. Comp. 73 (2004) 777–812) to check the convergence in L1(R) with rate O(1N+h) of the empirical cumulative distribution function of the Euler discretization with step h of the particle system to the solution of a one-dimensional viscous scalar conservation law. Last, we prove that the bias of this stochastic particle method behaves as O(1N+h). We provide numerical results which confirm our theoretical estimates.

Existence of solutions for some non-Fredholm integro-differential equations with mixed diffusion

We establish the existence in the sense of sequences of solutions for certain integro-differential type equations in two dimensions involving the normal diffusion in one direction and the anomalous diffusion in the other direction in H2(R2) via the fixed point technique. The elliptic equation contains a second order differential operator without the Fredholm property. It is proved that, under the reasonable technical conditions, the convergence in L1(R2) of the integral kernels implies the existence and convergence in H2(R2) of the solutions.

Existence in the Sense of Sequences of Stationary Solutions for Some Non-Fredholm Integro-Differential Equations

We establish the existence in the sense of sequences of stationary solutions for some reaction-diffusion type equations in appropriate H2 spaces. It is shown that, under reasonable technical conditions, the convergence in L1 of the integral kernels implies the existence and convergence in H2 of solutions. The nonlocal elliptic equations involve second order differential operators with and without the Fredholm property. Bibliography: 21 titles.

New characterizations of magnetic Sobolev spaces

Abstract We establish two new characterizations of magnetic Sobolev spaces for Lipschitz magnetic fields in terms of nonlocal functionals. The first one is related to the BBM formula, due to Bourgain, Brezis and Mironescu. The second one is related to the work of the first author on the classical Sobolev spaces. We also study the convergence almost everywhere and the convergence in L 1 {L^{1}} appearing naturally in these contexts.

Asymptotic properties of a stochastic chemostat including species death rate

This paper deals with a stochastic system which models the population dynamics of a chemostat including species death rate. On the basis of the theory on Markov semigroup, we demonstrate that the probability densities of the distributions for the solutions are absolutely continuous. The densities will convergence in L1 to an invariant density or weakly convergence to a singular measure under appropriate conditions. We also give the sufficient criteria for extinction exponentially of the species. To be specific, when D1>D and the strength of perturbation is relatively small, we derive a precise threshold for the species survival.

The high-order compact numerical algorithms for the two-dimensional fractional sub-diffusion equation

In this paper, performing the average operators on the space variables, a numerical scheme with third-order temporal convergence for the two-dimensional fractional sub-diffusion equation is considered, for which the unconditional stability and convergence in L1(L∞)-norm are strictly analyzed for α ∈ (0, 0.9569347] by using the discrete energy method. Therewith, adding small perturbation terms, we construct a compact alternating direction implicit difference scheme for the two-dimensional case. Finally, some numerical results have been given to show the computational efficiency and numerical accuracy of both schemes for all α ∈ (0, 1).

On Chung’s Law of Large Numbers on Simply Connected Step 2-Nilpotent Lie Groups

Ordonez Cabrera and Sung (2002) have proved that under certain “moment” conditions, for triangular arrays of weighted Banach-valued random variables, a.s. convergence, convergence in L 1, convergence in probability, and complete convergence to 0 are equivalent, thus giving a variant of Chung’s law of large numbers. We extend their result (under slightly sharper technical conditions) to symmetric random variables on simply connected step 2-nilpotent Lie groups.

Almost Everywhere Convergence of Convolution Measures

AbstractLet (X, ℬ, m, τ) be a dynamical system with (X, ℬ, m) a probability space and τ an invertible, measure preserving transformation. This paper deals with the almost everywhere convergence in L1(X) of a sequence of operators of weighted averages. Almost everywhere convergence follows once we obtain an appropriate maximal estimate and once we provide a dense class where convergence holds almost everywhere. The weights are given by convolution products of members of a sequence of probability measures {vi} defined on ℤ. We then exhibit cases of such averages where convergence fails.

The BGK Model with External Confining Potential: Existence, Long-Time Behaviour and Time-Periodic Maxwellian Equilibria

We study global existence and long time behaviour for the inhomogeneous nonlinear BGK model for the Boltzmann equation with an external confining potential. For an initial datum f 0≥0 with bounded mass, entropy and total energy we prove existence and strong convergence in L 1 to a Maxwellian equilibrium state, by compactness arguments and multipliers techniques. Of particular interest is the case with an isotropic harmonic potential, in which Boltzmann himself found infinitely many time-periodic Maxwellian steady states. This behaviour is shared with the Boltzmann equation and other kinetic models. For all these systems we study the multistability of the time-periodic Maxwellians and provide necessary conditions on f 0 to identify the equilibrium state, both in L 1 and in Lyapunov sense. Under further assumptions on f, these conditions become also sufficient for the identification of the equilibrium in L 1.

A NONLOCAL CONSERVATION LAW WITH NONLINEAR "RADIATION" INHOMOGENEITY

A scalar conservation law with a nonlinear dissipative inhomogeneity, which serves as a simplified model for nonlinear heat radiation effects in high-temperature gases is studied. Global existence and uniqueness of weak entropy solutions along with L1 contraction and monotonicity properties of the solution semigroup is established. Explicit threshold conditions ensuring formation of shocks within finite time is derived. The main result proves — under further assumptions on the nonlinearity and on the initial datum — large time convergence in L1 to the self-similar N-waves of the homogeneous conservation law.

The Singularity Property of Banach Function Spaces and Unconditional Convergence in L1[0, 1

Let X be a Banach function space, L∞ [0, 1] ⊂ X ⊂ L1[0, 1]. It is proved that if dual space of X has singularity property in closed set E ⊂ [0, 1] then: 1) there exists no orthonormal basis in C[0, 1], which forms an unconditional basis in X in metric of L1[0, 1] space, 2) for the Hardy-Littlewood maximal operator M we have Open image in new window

Intermediate Asymptotics Beyond Homogeneity and Self-Similarity: Long Time Behavior for u t = Δϕ(u)

We investigate the long time asymptotics in L1+(R) for solutions of general nonlinear diffusion equations u t = Δϕ(u). We describe, for the first time, the intermediate asymptotics for a very large class of non-homogeneous nonlinearities ϕ for which long time asymptotics cannot be characterized by self-similar solutions. Scaling the solutions by their own second moment (temperature in the kinetic theory language) we obtain a universal asymptotic profile characterized by fixed points of certain maps in probability measures spaces endowed with the Euclidean Wasserstein distance d2. In the particular case of ϕ(u) ~ u m at first order when u ~ 0, we also obtain an optimal rate of convergence in L1 towards the asymptotic profile identified, in this case, as the Barenblatt self-similar solution corresponding to the exponent m. This second result holds for a larger class of nonlinearities compared to results in the existing literature and is achieved by a variation of the entropy dissipation method in which the nonlinear filtration equation is considered as a perturbation of the porous medium equation.

CONSISTENCY OF ASYMMETRIC KERNEL DENSITY ESTIMATORS AND SMOOTHED HISTOGRAMS WITH APPLICATION TO INCOME DATA

We consider asymmetric kernel density estimators and smoothed histograms when the unknown probability density function f is defined on [0,+∞). Uniform weak consistency on each compact set in [0,+∞) is proved for these estimators when f is continuous on its support. Weak convergence in L1 is also established. We further prove that the asymmetric kernel density estimator and the smoothed histogram converge in probability to infinity at x = 0 when the density is unbounded at x = 0. Monte Carlo results and an empirical study of the shape of a highly skewed income distribution based on a large microdata set are finally provided.We thank O. Linton and the three referees for constructive criticism and M.P. Feser and J. Litchfield for kindly providing the Brazilian data. We are grateful to I. Gijbels, J.M. Rolin, and I. Van Keilegom for their stimulating remarks and to participants at the workshop on statistical modeling (UCL 2002), LAMES (Sao Paulo 2002), L1 Norm conference (Neuchatel 2002), Geneva econometrics seminar, and KUL econometrics seminar for their comments. Part of this research was done when the second author was visiting THEMA and IRES. The first, resp. second, author gratefully acknowledges financial support from the “Projet d'Actions de Recherche Concertées” grant 98/03-217, and from the IAP research network grant P5/24 of the Belgian state, resp. the Swiss National Science Foundation through the National Centre of Competence in Research: Financial Valuation and Risk Management (NCCR-FINRISK).

Entropy and Convergence on Compact Groups

We investigate the behaviour of the entropy of convolutions of independent random variables on compact groups. We provide an explicit exponential bound on the rate of convergence of entropy to its maximum. Equivalently, this proves convergence of the density to uniformity, in the sense of Kullback–Leibler. We prove that this convergence lies strictly between uniform convergence of densities (as investigated by Shlosman and Major), and weak convergence (the sense of the classical Ito–Kawada theorem). In fact it lies between convergence in L1+e and convergence in L1.

Edgeworth's conjecture with infinitely many commodities: commodity differentiation

Convergence of the cores of finite economies to the set of Walrasian allocations as the number of agents grows has long been taken as one of the basic tests of perfect competition. The present paper examines this test in the most natural model of commodity differentiation: the commodity space is the space of nonnegative measures, endowed with the topology of weak convergence. In Anderson and Zame [12], we gave counterexamples to core convergence in L 1, a space in which core convergence holds for replica economies and core equivalence holds for continuum economies; in addition, we gave a core convergence theorem under the assumption that traders' utility functions exhibit uniformly vanishing marginal utility at infinity. In this paper, we provide two core convergence results for the commodity differentiation model. A key technical virtue of this space is that relatively large sets (in particular, closed norm-bounded sets) are compact. This permits us to invoke a version of the Shapley-Folkman Theorem for compact subsets of an infinite-dimensional space. We show that, for sufficiently large economies in which endowments come from a norm bounded set, preferences satisfy an equidesirability condition, and either (i) preferences exhibit uniformly bounded marginal rates of substitution or (ii) endowments come from an order-bounded set, core allocations can be approximately decentralized by prices.

The Optimal Convergence Rate of Monotone Finite Difference Methods for Hyperbolic Conservation Laws

We are interested in the rate of convergence in L1 of the approximate solution of a conservation law generated by a monotone finite difference scheme. Kuznetsov has proved that this rate is 1/2 [USSR Comput. Math. Math. Phys., 16 (1976), pp. 105--119 and Topics Numer. Anal. III, in Proc. Roy. Irish Acad. Conf., Dublin, 1976, pp. 183--197], and recently Teng and Zhang have proved this estimate to be sharp for a linear flux [SIAM J. Numer. Anal., 34 (1997), pp. 959--978]. We prove, by constructing appropriate initial data for the Cauchy problem, that Kuznetsov's estimates are sharp for a nonlinear flux as well.

Banach space properties of $L\sp 1$ of a vector measure

We consider the space LI (v) of real functions which are integrable with respect to a measure v with values in a Banach space X. We study type and cotype for LI (v) . We study conditions on the measure v and the Banach space X that imply that LI (v) is a Hilbert space, or has the Dunford-Pettis property. We also consider weak convergence in L1 (v) .

An Inverse Sidon Type Inequality for the Walsh System

Upper estimations for the integral norm of the linear combinations of Dirichlet kernels expressed only in terms of the coefficients are called Sidon type inequalities. They can be applied in various fields of harmonic analysis. Among them are integrability and L1 convergence problems and summation methods. Several improvements of Sidon′s original inequality have been proved, most of them invariant with respect to the rearrangements of the coefficients, but no nontrivial lower estimation has been given so far. In this paper we give the best rearrangement invariant expression, which corresponds to an Orlicz norm, of coefficients for a Sidon type inequality. Examples for applications for convergence in L1 and in the dyadic Hardy space are also given. These questions are considered for the Walsh system.

L 1-convergence of double cosine- and Walsh-Fourier series

We prove the convergence inL 1([−gp, π)2)-norm of the double Fourier series of an integrable functionf(x, y) which is periodic and even with respect tox andy, with coefficientsa jk satisfying certain conditions of Hardy-Karamata kind, and such thata jk logj logk→0 asj, k→∞. These sufficient conditions become quite natural in particular cases. Then we extend these results to the convergence of double Walsh-Fourier series inL 1 (0, 1)2)- norm. As a by-product, we obtain Tauberian conditions ensuring the convergence of a double numerical series provided it is Cesaro summable.

Higher integrability of determinants and weak convergence in L1.

Higher integrability of determinants and weak convergence in L1.