Tauberian and Abelian Theorems for Long-range Dependent Random Fields

This paper surveys Abelian and Tauberian theorems for long-range dependent random fields. We describe a framework for asymptotic behaviour of covariance functions or variances of averaged functionals of random fields at infinity and spectral densities at zero. The use of the theorems and their limitations are demonstrated through applications to some new and less-known examples of covariance functions of long-range dependent random fields.     

Limit Theorems for Weighted Functionals of Cyclical Long-Range Dependent Random Fields

The article investigates isotropic random fields for which the spectral density is unbounded at some frequencies. Limit theorems for weighted functionals of these random fields are established. It is shown that for a wide class of functionals, which includes the Donsker scheme, the limit is not affected by singularities at non-zero frequencies. For general schemes, in contrast to the Donsker line, we demonstrate that the singularities at non-zero frequencies play a role even for linear functionals.     

Stationary self-similar random fields on the integer lattice

We establish several methods for constructing stationary self-similar random fields (ssf's) on the integer lattice by “random wavelet expansion”, which stands for representation of random fields by sums of randomly scaled and translated functions, or more generally, by composites of random functionals and deterministic wavelet expansion. To construct ssf's on the integer lattice, random wavelet expansion is applied to the indicator functions of unit cubes at integer sites. We demonstrate how to construct Gaussian, symmetric stable, and Poisson ssf's by random wavelet expansion with mother wavelets having compact support or non-compact support. We also generalize ssf's to stationary random fields which are invariant under independent scaling along different coordinate axes. Finally, we investigate the construction of ssf's by combining wavelet expansion and multiple stochastic integrals.     

On a Chen–Fliess approximation for diffusion functionals

We show that the so-called functional derivatives, as recently introduced by Dupire (Functional Ito calculus, SSRN, 2010), can provide intuitive meaning to classic expansions of path dependent functionals that appear in control theory (work of Brockett, Fliess, Sussmann et. al). We then focus on stochastic differential equations and show that vector fields can be lifted to act as derivations on such functionals. This allows to revisit and generalize the classic stochastic Taylor expansion to arrive at a Chen–Fliess approximation for smooth, path dependent functionals of SDEs with a corresponding \(L^{2}\) -error estimate.     

Joint distributions of random sets and their relation to copulas

Random sets are set-valued random variables. They have been applied in various fields like stochastic geometry, statistics, economics, engineering or computer science, and are often used for modeling uncertainty. In an earlier paper the author has defined joint capacity and joint containment functionals which are multivariate set functions describing the joint distribution of random sets. This paper is concerned with the question how copulas can be used to describe or model the dependence of random sets. It is demonstrated that a joint containment functional can be related to its margins by a family of copulas. Furthermore, the paper provides a first insight how copulas can be used to define joint containment functionals.     

Lorentz-Covariant Ultradistributions, Hyperfunctions, and Analytic Functionals

We generalize the theory of Lorentz-covariant distributions to broader classes of functionals including ultradistributions, hyperfunctions, and analytic functionals with a tempered growth. We prove that Lorentz-covariant functionals with essential singularities can be decomposed into polynomial covariants and establish the possibility of the invariant decomposition of their carrier cones. We describe the properties of odd highly singular generalized functions. These results are used to investigate the vacuum expectation values of nonlocal quantum fields with an arbitrary high-energy behavior and to extend the spin–statistics theorem to nonlocal field theory.     

Upper and lower functions of plane Brownian angles

One of the most rapidly growing fields in the theory of random processes is asymptotics of multidimensional Wiener functionals, including such functionals as winding angles or radius-vectors of planar processes. We present a new method yielding a large number of results on upper and lower functions for such functionals. Bibliography: 23 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 228, 1996, pp. 111–134.     

A random point field related to Bose-Einstein condensation

The random point fields which describe the position distributions of the systems of ideal boson gas in states of Bose-Einstein condensations are obtained through the thermodynamic limits. The resulting point fields are given by convolutions of two kinds of independent point fields: the so-called boson processes whose generating functionals are represented by the inverses of the Fredholm determinants for operators related to the heat operator and the point fields whose generating functionals are represented by the resolvents of the operators. The construction of the latter point fields in an abstract formulation is also given.     

Equivalence between the Lyapunov–Krasovskii functionals approach for discrete delay systems and that of the stability conditions for switched systems

The stability of discrete-time systems with time varying delay in the state can be analyzed by using a discrete-time extension of the classical Lyapunov–Krasovskii approach. In the networked control systems domain a similar delay stability problem is treated using a switched system transformation approach. The paper aims to establish a relation between the switched system transformation approach and the classical Lyapunov–Krasovskii method. It is shown that using the switched systems transformation is equivalent to using a general delay dependent Lyapunov–Krasovskii functionals. This functional represents the most general form that can be obtained using sums of quadratic terms. Necessary and sufficient LMI conditions for the existence of such functionals are presented.     

Continuous Essential Selections and Integral Functionals

Given a strictly positive measure, we characterize inner semicontinuous solid convex-valued mappings for which continuous functions which are selections almost everywhere are selections. This class contains continuous mappings as well as fully lower semicontinuous closed convex-valued mappings that arise in variational analysis and optimization of integral functionals. The characterization allows for extending existing results on convex conjugates of integral functionals on continuous functions. We also give an application to integral functionals on left continuous functions of bounded variation.     

Tauberian and Abelian theorems for correlation function of a homogeneous isotropic random field

We prove theorems of Tauberian and Abelian types for nonintegrable correlation functions of homogeneous isotropic random fields and use them to study asymptotic distributions of local functionals of Gaussian fields.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 12, pp. 1652–1664, December, 1991.     

Diffraction tomography: Construction and interpretation of tomographic functionals

On the basis of a linearized model of propagation of seismic wave fields the notion of tomographic functionals is introduced. The physical interpretation of tomographic functionals is that their integral kernels are spatial functions of the influence of variations of the medium parameters in question upon particular measurements of the wave field of the sounding signal. The norm of a tomographic functional is determined by the intensity of the influence function related to the interaction operator. The field is generated by a “source” with the dependence on time determined by the apparatus function of the seismic channel. The analysis of tomographic functionals makes it possible to mathematically design tomographic experiments for monitoring active seismic zones by controlling the parameters of tomographic functionals. The richness in content of a tomographic experiments is determined not only by the norms of tomographic functionals, but also by the region where their supports overlap. Tomographic functionals for the wave and Lamé equations are analyzed. Bibliography:14 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 218, 1994, pp. 176–196. This work was supported by the Russian Foundation of Fundamental Research (Grant 93059961) and by the ISF. Translated by V. N. Troyan.     

On Γ-convergence in divergence-free fields through Young measures

The explicit characterization of the limit energy density of sequences of general non-periodic functionals defined on divergence-free fields is achieved by means of the div-Young measure associated with relevant sequences of functions through a minimization problem.     

Limit distributions of certain functionals of homogeneous isotropic Gaussian fields with strong dependency

One considers limit theorems for functionals of geometric type, generated by the intersections of the realizations of Gaussian homogeneous random fields with planes and by radial functions. Special attention is given to fields where the integral of the correlation function is divergent.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 350–372, 1986.     

Localization of limit sets of solutions of nonautonomous delay equations by nonmonotone functionals

We present new generalizations of the Barbashin-Krasovskii theorem which also apply to delay equations with unbounded right-hand side. These generalizations are based on information on the localization of the limit sets of solutions, which is obtained with the use of two classes of not necessarily monotone Lyapunov functionals. The classes of functionals to be used contain sign-definite Lyapunov-Krasovskii functionals as well as Lyapunov-Razumikhin functions.     

Construction of functions with uniform sublevel sets

In this paper, we study extended real-valued functions with uniform sublevel sets. The sublevel sets are defined by a linear shift of a set in a specified direction. We prove that the class of these functions coincides with the class of Gerstewitz functionals. In this way, we obtain a formula for the construction of such functions. The sublevel sets of Gerstewitz functionals are characterized and illustrated by examples. The results contain statements for translative functions, which are just the functions with uniform sublevel sets considered. The investigated functions are defined on an arbitrary real vector space without assuming any topology or convexity.     

Almost sure central limit theorems on the Wiener space

In this paper, we study almost sure central limit theorems for sequences of functionals of general Gaussian fields. We apply our result to non-linear functions of stationary Gaussian sequences. We obtain almost sure central limit theorems for these non-linear functions when they converge in law to a normal distribution.     

Entropy method for line-energies

In this paper we analyze energy functionals concentrated on the discontinuity lines of unit-length, divergence-free 2D vector fields. The motivation comes from thin-film micromagnetics where these functionals correspond to mesoscopic wall-energies. A natural issue consists in characterizing the line-energy densities for which the functionals are lower semicontinuous for a relevant topology. In fact, this is a necessary condition for being the Γ-limit of a family of energies. A key point in our study is the use of the notion of entropy production borrowed from the field of conservation laws. With this tool, we build a large class of lower semicontinuous line-energies. In particular, we prove that certain power functions lead to such line-energy functionals as conjectured in Ambrosio et?al. (Calc Var Partial Differ Equ 9(4):327–355, 1999). We also deduce compactness properties for these functionals leading to the existence of minimizers for their lower semicontinuous envelopes. Another natural question is whether the viscosity solution is a minimizing configuration. We show that the answer is in general negative by exhibiting some special nonconvex domains as counterexamples. However, we establish positive results for some special domains (stadium, ellipse and union of two discs). The case of general convex domains is still open.     

Tractability of Tensor Product Linear Operators

This paper deals with the worst case setting for approximating multivariate tensor product linear operators defined over Hilbert spaces. Approximations are obtained by using a number of linear functionals from a given class of information. We consider the three classes of information: the class of all linear functionals, the Fourier class of inner products with respect to given orthonormal elements, and the standard class of function values. We wish to determine which problems are tractable and which are strongly tractable. The complete analysis is provided for approximating operators of rank two or more. The problem of approximating linear functionals is fully analyzed in the first two classes of information. For the third class of standard information we show that the possibilities are very rich. We prove that tractability of linear functionals depends on the given space of functions. For some spaces all nontrivial normed linear functionals are intractable, whereas for other spaces all linear functionals are tractable. In “typical” function spaces, some linear functionals are tractable and some others are not.     

Extension of the variational formulation of the problem for a rigid-plastic medium to velocity fields with slip-type discontinuities

Sets of velocity fields containing slip-type discontinuities at the boundary of the rigid-plastic medium, as well as within it, and the functionals defined on these sets, are described. It is shown that the exact lower bounds of the variational problems for these functionals are equal to the coefficient of the critical load. The minimax problem with saddle point constructed here is regarded as an extension of the classical minimax problem of the theory of critical loads.