Young measures generated by sequences in Morrey spaces

Let ${\Omega\subset\mathbb{R}^n}$ be open and bounded. For 1 ≤ p < ∞ and 0 ≤ λ < n, we give a characterization of Young measures generated by sequences of functions ${\{{\bf f}_j\}_{j=1}^\infty}$ uniformly bounded in the Morrey space ${L^{p,\lambda}(\Omega;\mathbb{R}^N)}$ with ${\{\left|{{\bf f}_j}\right|^p\}_{j=1}^\infty}$ equiintegrable. We then treat the case that each f _j = ? u _j for some ${{\bf u}_j\in W^{1,p}(\Omega;\mathbb{R}^N)}$ . As an application of our results, we consider the functional $${\bf u} \mapsto \int\limits_{\Omega}f({\bf x}, {\bf u}({\bf x}), {\bf {\nabla}}{\bf u}({\bf x})){\rm d}{\bf x},$$ and provide conditions that guarantee the existence of a minimizing sequence with gradients uniformly bounded in ${L^{p,\lambda}(\Omega;\mathbb{R}^{N\times n})}$ .     

Gradient Young measures generated by sequences in Sobolev spaces

Oscillatory properties of a weak convergent sequence of functions bounded inL ^p , 1 ≤p ≤ ∞, may be summarized by the parametrized measure it generates. When such a measure is generated by the gradients of a sequence of functions bounded inH ^1,p , it must have special properties. The purpose of this paper is to characterize such parametrized measures as the ones that obey Jensen’s inequality for all quasiconvex functions with the appropriate growth at infinity. We have found subtle differences between the casesp < ∞ andp = ∞. A consequence is that any measure determined by biting convergence is in fact generated by a sequence convergent in a stronger sense. We also give a few applications. Research groupTransitions and Defects in Ordered Materials, funded by the NSF, the AFOSR, and the ARO. The work of the second author is also supported by DGICYT (Spain) through “Programa de Perfeccionamiento y Movilidad del Personal Investigador” and through Grant PB90-0245.     

An open mapping theorem for Young measures

Ditor and Eifler consider the open mapping theorem for the probability spaces. Here we attempt to generalize the theorem to the spaces of Young measures.

     

A Fatou lemma for Gelfand integrals by Means of Young Measure Theory

It is shown that a Fatou-type lemma for Gelfand-integrable functions, due to Cornet and Medecin, can be generalized by following the approach to Fatou-type results introduced in (Balder, Math. Op. Res. 9 (1984), 270; Balder, J. Math. Anal. Appl. 136 (1988), 450)     

Young measure approach to characterization of behaviour of integral functionals on weakly convergent sequences by means of their integrands

It is known that sequential weak lower semicontinuity and weak-strong convergence (in the scalar case) properties of integral functionals may be characterized by means of their integrands. In this paper we introduce a Young measure approach obtaining both these results and the characterization for the second property in the vector-valued case. We discuss also motivations for the definition of strict quasiconvexity, and point out that the characterization of the classes of functionals having weak-strong convergence property everywhere is not a trivial problem in the general case.     

Young measures generated by a class of integrands: a narrow epicontinuity and applications to Homogenization

Given a subset E of convex functions from into which satisfy growth conditions of order p>1 and an open bounded subset of , we establish the continuity of a map μΦ_μ from the set of all Young measures on equipped with the narrow topology into a set of suitable functionals defined in and equipped with the topology of Γ-convergence. Some applications are given in the setting of periodic and stochastic homogenization.     

Description of bilinear forms generated by indefinite vector measures

We describe a correlation function generated by a J-orthogonal indefinite measure with values in a Krein space.     

A q-Analog of the Hook Walk Algorithm for Random Young Tableaux

A probabilistic algorithm, called the q-hook walk, is defined. For a given Young diagram, it produces a new one by adding a random box with probabilities, depending on a positive parameter q. The corresponding Markov chain in the space of infinite Young tableaux is closely related to the knot invariant of Jones, constructed via traces of Hecke algebras. For q = 1, the algorithm is essentially the hook walk of Greene, Nijenhuis, and Wilf. The q-hook formula and a q-deformation of Young graph are also considered.S. Kerov: Supported by a grant from CRM (Université de Montréal), during its Operator Algebras year     

The influence of deterministic noise on empirical measures generated by stationary processes

We consider weak convergence of empirical measures generated by stationary random process perturbed by deterministic noise . We assume that the noise has asymptotic distribution. In particular, we demonstrate that if the process is ergodic, or satisfies some mixing assumptions, then the influence of deterministic noise on is the same as it would be if were stochastic. Such results are of importance when investigating fluctuations and convex rearrangements of stochastic processes.

     

Young measure solutions and instability of the one-dimensional Perona-Malik equation

We apply a variational approach to the one-dimensional version of the widely used Perona-Malik equation in image processing. We rephrase the problem into the one related to the quasiconvex hull of a graph in the space of 2×2 matrices M2×2. We then use the solutions of some heat equations as the centre of the mass for the Young measure-valued solutions to construct the approximate solutions by using simple laminates. The approximate solutions can be viewed as solutions of a perturbation problem by W−1,p (or W−1,∞) functions. The sequences of the approximate solutions generates Young measure-valued solutions. Our results also show that the solutions of the one-dimensional Perona-Malik equation are unstable under small W−1,∞ perturbations.     

A decomposition for the exponential dispersion model generated by the invariant measure on the hyperboloid

For = ₀, ₁, ₂) andx=(x₀, x₁, x₂) in R³, define [,x] = ₀ x ₀ – ₁ x ₁ – ₂ x ₂,C = {x³:x ₀ > 0 and [x, x]>0},R(x)=([x, x]) ^1/2 forx inC andH ₁={xC: x₀>0,R(x)=1}. Define the measure onH ₁ such that if is inC and =R(), then exp (–[,x])(dx = ( exp )^–1. Therefore, is invariant under the action ofSO (1, 2), the connected component ofO(1, 2) containing the identity. We first prove that there exists a positive measure in ³ such that its Laplace transform is ( exp )^– if and only if >1. Finally, for 1 and inC, denotingP(,)(dx) = ( exp )^– exp (–[,x])(dx, we show that ifY ₀,...,Y _n aren+1 independent variables with densityP(,),j=0,...,n and ifS _k =X ₀ + ... +X _k andQ _k =R(S _k) –R(S _k–1) –R(Y _k),k=1,...,n, then then+1 statisticsD _n = [/,S _k ] –R _{k – 1} ),Q ₁,...,Q _n are independent random variables with the exponential () or gamma (1,1/) distribution.This research has been partially funded by NSERC Grant A8947.     

An elementary method of calculating an explicit form of Young measures in some special cases

We present an elementary method of explicit calculation of Young measures for a certain class of functions. This class contains, in particular, functions of a highly oscillatory nature which appear in optimization problems and homogenization theory. In engineering such situation occurs, for instance, in nonlinear elasticity (solid–solid phase transition in certain elastic crystals). Young measures associated with oscillating minimizing sequences gather information about their oscillatory nature and therefore about underlying microstructure. The method presented in this article makes no use of functional analytic tools. There is no need to use a generalized version of the Riemann–Lebesgue lemma and to calculate weak* limits of functions. The main tool is the change of variable theorem. The method applies both to sequences of periodic and nonperiodic functions.     

A combinatorial proof of the skew K-saturation theorem

We give a combinatorial proof of the skew version of the K-saturation theorem. More precisely, for any positive integer k$k$, we give an explicit injection from the set of skew semistandard Young tableaux with skew shape kλ/kμ$k\lambda /k\mu$ and type kν$k\nu$, to the set of skew semistandard Young tableaux of shape λ/μ$\lambda /\mu$ and type ν$\nu$.     

A combinatorial proof of the log-convexity of sequences in Riordan arrays

Journal of Algebraic Combinatorics - A Riordan array $$R=[r_{n,k}]_{n,k\ge 0}$$ can be characterized by two sequences $$A=(a_n)_{n\ge 0}$$ and $$Z=(z_n)_{n\ge 0}$$ such that $$r_{0,0}=1,...     

A limit theorem for sequences generated by Weyl transformation: Disappearance of dependence

 We prove a limit theorem for certain pseudo-random sequences generated by Weyl transformations, thereby extending a result of Sugita in [3]. Received: 8 June 2001 / Revised version: 4 March 2002 / Published online: 22 August 2002     

A remark on an integral characterization of the dual of BV

In this paper, we show how under the continuum hypothesis one can obtain an integral representation for elements of the topological dual of the space of functions of bounded variation in terms of Lebesgue and Kolmogorov–Burkill integrals.     

Convergence of a misanthrope process to the entropy solution of 1D problems

We prove the convergence, in some strong sense, of a Markov process called “a misanthrope process” to the entropy weak solution of a one-dimensional scalar nonlinear hyperbolic equation. Such a process may be used for the simulation of traffic flows. The convergence proof relies on the uniqueness of entropy Young measure solutions to the nonlinear hyperbolic equation, which holds for both the bounded and the unbounded cases. In the unbounded case, we also prove an error estimate. Finally, numerical results show how this convergence result may be understood in practical cases.     

A new proof for the conditions of Novikov and Kazamaki

This paper provides a novel proof for the sufficiency of certain well-known criteria that guarantee the martingale property of a continuous, nonnegative local martingale. More precisely, it is shown that generalizations of Novikov’s condition and Kazamaki’s criterion follow directly from the existence of Föllmer’s measure. This approach allows to extend well-known criteria of martingality from strictly positive to only nonnegative, continuous local martingales.