Limiting behaviour for arrays of row-wise END random variables under conditions of h-integrability

The authors study L^r-convergence, complete convergence and complete moment convergence for arrays of row-wise extended negatively dependent random variables under some appropriate conditions of h-integrability. The results in this paper extend and improve the results of Sung et al. [H.S. Sung, S. Lisawadi, A. Volodin, Weak laws of large numbers for arrays under a condition of uniform integrability, J. Korean Math. Soc. 45 (2008), pp. 289–300].     

Homogenization of problems of elasticity theory on composite structures with thin armoring

The paper considers the problems of elasticity theory on a flat slab armored by a periodic thin mesh or in a three-dimensional body armored by a periodic thin box structure. The composite medium depends on two small mutually related geometric parameters; one of them controls the periodicity cell and the other controls the thickness of the armoring structure. It is proved that the homogenization of the indicated problems is classical. In doing so, one applies V. V. Zhikov’s approach (“Zhikov measure approach”) together with the two-scale convergence method. Preliminarily, the paper studies the peculiarities of the two-scale convergence with the variable composite measure and also the Sobolev spaces of elasticity theory with variable composite measure. The obtained compactness principle (an analog of the Rellich theorem) in these spaces made it possible to prove the Hausdor. convergence of the spectrum of the problem studied. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical Systems and Optimization, 2005.     

Two-scale convergence of some integral functionals

Nguetseng’s notion of two-scale convergence is reviewed, and some related properties of integral functionals are derived. The coupling of two-scale convergence with convexity and monotonicity is then investigated, and a two-scale version is provided for compactness by strict convexity. The div-curl lemma of Murat and Tartar is also extended to two-scale convergence, and applications are outlined.     

On two-scale convergence and related sequential compactness topics

A general concept of two-scale convergence is introduced and two-scale compactness theorems are stated and proved for some classes of sequences of bounded functions in L ²(Ω) involving no periodicity assumptions. Further, the relation to the classical notion of compensated compactness and the recent concepts of two-scale compensated compactness and unfolding is discussed and a defect measure for two-scale convergence is introduced.     

Random Function Iterations for Consistent Stochastic Feasibility

We study the convergence of iterated random functions for stochastic feasibility in the consistent case (in the sense of Butnariu and Flåm [Numer. Funct. Anal. Optimiz., 1995]) in several different settings, under decreasingly restrictive regularity assumptions of the fixed point mappings. The iterations are Markov chains and, for the purposes of this study, convergence is understood in very restrictive terms. We show that sufficient conditions for geometric (linear) convergence in expectation of stochastic projection algorithms presented in Nedi? [Math. Program, 2011], are in fact necessary for geometric (linear) convergence in expectation more generally of iterated random functions.     

Uncertainty quantification for random parabolic equations with nonhomogeneous boundary conditions on a bounded domain via the approximation of the probability density function

This paper deals with the randomized heat equation defined on a general bounded interval [L₁, L₂] and with nonhomogeneous boundary conditions. The solution is a stochastic process that can be related, via changes of variable, with the solution stochastic process of the random heat equation defined on [0,1] with homogeneous boundary conditions. Results in the extant literature establish conditions under which the probability density function of the solution process to the random heat equation on [0,1] with homogeneous boundary conditions can be approximated. Via the changes of variable and the Random Variable Transformation technique, we set mild conditions under which the probability density function of the solution process to the random heat equation on a general bounded interval [L₁, L₂] and with nonhomogeneous boundary conditions can be approximated uniformly or pointwise. Furthermore, we provide sufficient conditions in order that the expectation and the variance of the solution stochastic process can be computed from the proposed approximations of the probability density function. Numerical examples are performed in the case that the initial condition process has a certain Karhunen‐Loève expansion, being Gaussian and non‐Gaussian.     

Rates of convergence in the CLT for linear random fields

In this paper, we study sums of linear random fields defined on the lattice Z ² with values in a Hilbert space. The rate of convergence of distributions of such sums to the Gaussian law is discussed, and mild sufficient conditions to obtain an approximation of order n ^−p are presented. This can be considered as a complement of a recent result of [A.N. Nazarova, Logarithmic velocity of convergence in CLT for stochastic linear processes and fields in a Hilbert space, Fundam. Prikl. Mat., 8:1091–1098, 2002 (in Russian)], where the logarithmic rate of convergence was stated, and as a generalization of the result of [D. Bosq, Erratum and complements to Berry–Esseen inequality for linear processes in Hilbert spaces, Stat. Probab. Lett., 70:171–174, 2004] for linear processes.     

On Two-Scale Convergence

In this paper we present the foundations of the two-scale convergence theory. We discuss the class of admissible functions in the mean-value formula and in the definition of two-scale convergence. We discuss the relation between strong and weak convergence and prove general theorems on the semicontinuity from below for convex functionals, and we also discuss the relation between two-scale convergence and monotonicity. The explanations are conducted on the level of L ^p -convergence; we do not deal with convergence in Sobolev spaces.     

On approximative properties for thin structures

In this paper, we study approximative properties connecting a periodic Borel measure µ^h with its weak-limit measure µ as h → 0. V. V. Zhikov showed that these properties are necessary for averaging problems with two small parameters arising in thin periodic structures with thickness tending to zero.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 9, Suzdal Conference-3, 2003.     

Asymptotic results for weighted means of linear combinations of independent Poisson random variables

ABSTRACT

In this paper we prove the large deviation principle for a class of weighted means of linear combinations of independent Poisson distributed random variables, which converge weakly to a normal distribution. The interest in these linear combinations is motivated by the diffusion approximation in Lansky [On approximations of Stein's neuronal model, J. Theoret. Biol. 107 (1984), pp. 631–647] of the Stein's neuronal model (see Stein [A theoretical analysis of neuronal variability, Biophys. J. 5 (1965), pp. 173–194]). We also prove an analogue result for sequences of multivariate random variables based on the diffusion approximation in Tamborrino, Sacerdote, and Jacobsen [Weak convergence of marked point processes generated by crossings of multivariate jump processes. Applications to neural network modeling, Phys. D 288 (2014), pp. 45–52]. The weighted means studied in this paper generalize the logarithmic means. We also investigate moderate deviations.     

Feynman–Kac formulas for regime-switching jump diffusions and their applications

This work develops Feynman–Kac formulas for a class of regime-switching jump diffusion processes, in which the jump part is driven by a Poisson random measure associated with a general Lévy process and the switching part depends on the jump diffusion processes. Under broad conditions, the connections of such stochastic processes and the corresponding partial integro-differential equations are established. Related initial, terminal and boundary value problems are also treated. Moreover, based on weak convergence of probability measures, it is demonstrated that a sequence of random variables related to the regime-switching jump diffusion process converges in distribution to the arcsine law.     

Stochastic integration with respect to compensated Poisson random measures on separable Banach spaces

We define stochastic integrals of Banach valued random functions w.r.t. compensated Poisson random measures. Different notions of stochastic integrals are introduced and sufficient conditions for their existence are established. These generalize, for the case where integration is performed w.r.t. compensated Poisson random measures, the notion of stochastic integrals of real valued random functions introduced in Ikeda and Watanabe (1989) [Stochastic Differential Equations and Diffusion Processes (second edition), North-Holland Mathematical Library, Vol. 24, North Holland Publishing Company, Amsterdam/Oxford/New York.], (in a different way) in Bensoussan and Lions (1982) [Contróle impulsionnel et inquations quasi variationnelles. (French) [Impulse control and quasivariational inequalities] Méthodes Mathématiques de l'Informatique [Mathematical Methods of Information Science], Vol. 11. (Gauthier-Villars, Paris), and Skorohod, A.V. (1965) [Studies in the theory of random processes (Addison-Wesley Publishing Company, Inc, Reading, MA), Translated from the Russian by Scripta Technica, Inc. ], to the case of Banach valued random functions. The relation between these two different notions of stochastic integrals is also discussed here.     

Existence of Random Solutions of a General Class of Stochastic Functional Integral Equations

Abstract

In this paper, we establish the existence of solutions of a more general class of stochastic functional integral equations. The main tools here are the measure of noncompactness and the fixed point theorem of Darbo type. The results of this paper generalize the results of Rao–Tsokos [Rao, A.N.V.; Tsokos, C.P. A class of stochastic functional integral equations. Coll. Math. 1976, 35, 141–146.] and Szynal–Wedrychowicz [Szynal, D.; Wedrychowicz, S. On existence and an asymptotic behaviour of random solutions of a class of stochastic functional integral equations. Coll. Math. 1987, 51, 349–364.].     

Convergence of nonhomogeneous stochastic chains with countable states

The work started by V. M. Maksimov [1970, Theory Probab. Appl.15, 604–618], and continued by A. Mukherjea [1980, Trans. Amer. Math. Soc.263, 505–520], is extended, and completed with respect to certain aspects. Infinite-dimensional stochastic chains are considered in the framework of Mukherjea [loc. cit.]; backward products of stochastic matrices and their convergence are also considered. The main theme centers around understanding how the convergence of products (backward and forward, finite and infinite dimensional) takes place and what it means in terms of various types of asymptotic behavior of the individual stochastic matrices in the chain. The study is based on establishing the existence of a basis for convergent chains. The basis then makes it possible to describe properly various aspects of convergence. All results are new; they are also complete at least in the sense they have been presented and suitable examples (or counter-examples) are presented to justify the assumptions involved.     

Imbedded construction of stationary sequences and point processes with a random memory

We show convergence in variation to a unique stationary state for a class of point processes (respectively, stochastic sequences) with stochastic intensity kernels (respectively, transition probabilities) including the (A,m)-processes of Lindvall [12]. This is done under two basic conditions: first, the random memory of the processes considered is consistent or non-reusable (that is, past information not used at a given time cannot be recalled at a later time) and secondly, the kernels have a deterministic fixed component for which the memory is almost surely finite.     

Random Dirichlet problem: Scalar darcy's law

Using Ergodic Theory and Epiconvergence notion, we study the rate of convergence of solutions relative to random Dirichlet problems in domains ofR ^d with random holes whose size tends to 0. This stochastic analysis allows to extend the results already obtained in the corresponding periodic case.     

On the location of the maximum of a process: Lévy,Gaussian and Random field cases

ABSTRACT

We show how the techniques presented in Pimentel [On the location of the maximum of a continuous stochastic process, J. Appl. Prob. 51 (2014), pp. 152–161] can be extended to a variety of non-continuous processes and random fields. For the Gaussian case, we prove new covariance formulae between the maximum and the maximizer of the process. As examples, we prove uniqueness of the location of the maximum for spectrally positive Lévy processes, Ornstein–Uhlenbeck process, fractional Brownian Motion and the Brownian sheet among other processes.     

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.     

On non-Markovian forward–backward SDEs and backward stochastic PDEs

In this paper, we establish an equivalence relationship between the wellposedness of forward–backward SDEs (FBSDEs) with random coefficients and that of backward stochastic PDEs (BSPDEs). Using the notion of the “decoupling random field”, originally observed in the well-known Four Step Scheme (Ma et al., 1994 [13]) and recently elaborated by Ma et al. (2010) [14], we show that, under certain conditions, the FBSDE is wellposed if and only if this random field is a Sobolev solution to a degenerate quasilinear BSPDE, extending the existing non-linear Feynman–Kac formula to the random coefficient case. Some further properties of the BSPDEs, such as comparison theorem and stability, will also be discussed.     

Homogenized Diffusion Limit of a Vlasov–Poisson–Fokker–Planck Model

The approximation by diffusion and homogenization of the initial-boundary value problem of the Vlasov–Poisson–Fokker–Planck model is studied for a given velocity field with spatial macroscopic and microscopic variations. The L¹-contraction property of the Fokker–Planck operator and a two-scale Hybrid-Hilbert expansion are used to prove the convergence towards a homogenized Drift–Diffusion equation and to exhibit a rate of convergence.