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The existence and the upper semicontinuity of random attractors in L 2 ( R N ) for stochastic semi-linear degenerate parabolic equations with multiplicative noises are obtained. Though the spatial domain is the entire space R N , a compact embedding is applied to get the compactness of random absorbing sets, from which follows the compactness of the system and the precompactness of the union of all perturbed random attractors.

We obtain some random fixed point theorems for random mappings. We use the orbits of the random mappings to show the existence of a fixed point for a class of random mappings and also establish the measurability of solutions obtained through such random mappings. Some applications of these theorems to random integral equations are given.

We investigated a linear random dynamical system which strongly preserves a cone C of dimension- k (abbr. k -cone) in R n . Under some general assumptions, it is shown that such system admits a measurable family of k -dimensional subspaces and a measurable family of ( n − k ) -dimensional subspaces which are complementary to each other and form into a tempered invariant splitting of R n . We further apply the measurable bundles so obtained to study the linear random monotone cyclic feedback systems, as well as the linear competitive–cooperative tridiagonal systems. This generalizes the Floquet theory for these deterministic non-autonomous (or time-periodic) systems to the random systems.

A stochastic forcing of a non-linear singular/degenerated parabolic problem with random growth conditions is proposed in the framework of Orlicz Lebesgue and Sobolev spaces with variable random exponents. We give a result of existence and uniqueness of the solution, for additive and multiplicative problems.

Under the second moment condition, we obtain Berry-Esseen bounds for random index non linear statistics by using a technique discussed in Chen and Shao (2007). A concept in this article is to approximate any random index non-linear statistic by a random index linear statistic. The bounds for random sums of independent random variables are also provided. Applications are the bounds for random U-statistics and random sums of the present values in investment analysis.

We first prove the existence of random attractors for the continuous random dynamical systems generated by stochastic damped non-autonomous wave equations with linear memory and additive white noise when the nonlinearity has a critically growing exponent. Then we study the upper semicontinuity of random attractors when the coefficient of random term approaches zero.

In this note we consider the n × n random matrices whose ( i , j ) th entry is f ( x i T x j ) , where x i ’s are i.i.d. random vectors in R N , and f is a real-valued function. The empirical spectral distributions of these random inner-product kernel matrices are studied in two kinds of high-dimensional regimes: n / N → γ ∈ ( 0 , ∞ ) and n / N → 0 as both n and N go to infinity. We obtain the limiting spectral distributions for those matrices from different random vectors in R N including the points l p -norm uniformly distributed over four manifolds. And we also show a result on isotropic and log-concave distributed random vectors, which confirms a conjecture by Do and Vu.

This is a survey of known results related to the asymptotic behaviour of the probabilities of first-order properties of random graphs. The results presented in this paper are concerned with zero-one laws for properties of random graphs. Emphasis is placed on the Erdős-Renyi model of a random graph. Also considered are some generalizations of this model motivated by various problems in the theory of coding and combinatorial geometry. Bibliography: 65 titles.

This article addresses the following question; 'how to approximate the spectrum of random bounded self-adjoint operators on separable Hilbert spaces'. This is an attempt to establish a link between the spectral theory of random operators and the rich theory of random matrices; including various notions of convergence. This study tries to develop a random version of the truncation method, which is useful in approximating spectrum of bounded self- adjoint operators. It is proved that the eigenvalue sequences of the truncations converge in distribution to the eigenvalues of the random bounded self-adjoint operator. The convergence of moments are also proved with some examples. In addition, the article discusses some new methods to predict the existence of spectral gaps between the bounds of essential spectrum. Some important open problems are also stated at the end.

A concept of a bi-spatial random attractor for a random dynamical system is introduced. A unified result about existence and upper semi-continuity for a family of bi-spatial random attractors is obtained if a family of random systems is convergent, uniformly absorbing in an initial space and uniformly omega-compact in both initial and terminate spaces. The upper semi-continuity result improves all existing results even for single-spatial attractors. As an application of the abstract result, it is shown that every semilinear Laplacian equation on the entire space perturbed by a multiplicative and stochastic noise possesses an ( L 2 , L q ) -random attractor with q > 2 . Moreover, it is proved that the family of obtained attractors is upper semi-continuous at any density of noises and the family of attractors for the corresponding compact systems is both upper and lower semi-continuous at infinity under the topology of both spaces.

We study algorithmically random closed subsets of $2^\omega$, algorithmically random continuous functions from $2^\omega$ to $2^\omega$, and algorithmically random Borel probability measures on $2^\omega$, especially the interplay between these three classes of objects. Our main tools are preservation of randomness and its converse, the no randomness ex nihilo principle, which say together that given an almost-everywhere defined computable map between an effectively compact probability space and an effective Polish space, a real is Martin-L\"of random for the pushforward measure if and only if its preimage is random with respect to the measure on the domain. These tools allow us to prove new facts, some of which answer previously open questions, and reprove some known results more simply. Our main results are the following. First we answer an open question of Barmapalias, Brodhead, Cenzer, Remmel, and Weber by showing that $\mathcal{X}\subseteq2^\omega$ is a random closed set if and only if it is the set of zeros of a random continuous function on $2^\omega$. As a corollary we obtain the result that the collection of random continuous functions on $2^\omega$ is not closed under composition. Next, we construct a computable measure $Q$ on the space of measures on $2^\omega$ such that $\mathcal{X}\subseteq2^\omega$ is a random closed set if and only if $\mathcal{X}$ is the support of a $Q$-random measure. We also establish a correspondence between random closed sets and the random measures studied by Culver in previous work. Lastly, we study the ranges of random continuous functions, showing that the Lebesgue measure of the range of a random continuous function is always contained in $(0,1)$.

The Wiener index has been studied for simply generated random trees, nonplane unlabeled random trees, and a huge subclass of random grid trees containing random binary search trees, random median-of-(2k+1) search trees, random $m$-ary search trees, random quadtrees, random simplex trees, etc. An important class of random grid trees for which the Wiener index was not studied so far is random digital trees. In this work, we close this gap. More precisely, we derive asymptotic expansions of moments of the Wiener index and show that a central limit law for the Wiener index holds. These results are obtained for digital search trees and bucket versions as well as tries and PATRICIA tries. Our findings answer in the affirmative two questions posed by Neininger.

In this paper, we shall represent a strong law of large numbers (SLLN) for weighted sums of set- valued random variables in the sense of the Hausdorff metric dH, based on the result of single-valued random variable obtained by Taylor [1].

A formalism for the study of random dynamical systems with inputs and outputs (RDSIO) is introduced. An axiomatic framework and basic properties of RDSIO are developed, and a theorem is shown that guarantees the stability of interconnected systems.

International audience; In this paper, we characterize random eigenspaces with a non-intrusive method based on the decoupling of random eigenvalues from their corresponding random eigenvectors. This method allows us to estimate the first statistical moments of the random eigenvalues of the system with a reduced number of deterministic finite element computations. The originality of this work is to adapt the method used to estimate each random eigenvalue depending on a global accuracy requirement. This allows us to ensure a minimal computational cost. The stochastic model of the structure is thus reduced by exploiting specific properties of random eigenvectors associated with the random eigenfrequencies being sought. An indicator with no additional computation cost is proposed to identify when the method needs to be enhanced. Finally, a simple three-beam frame and an industrial structure illustrate the proposed approach.

In [17], the authors have defined an annealed Ising model on random graphs and proved limit theorems for the magnetization of this model on some random graphs including random 2-regular graphs. Then in [11], we generalized their results to the class of all random regular graphs. In this paper, we study the critical behavior of this model. In particular, we determine the critical exponents and prove a non standard limit theorem that the magnetization scaled by n 3/4 converges to a specific random variable, with n the number of vertices of random regular graphs., 23 pages, accepted for publication in Journal of Statistical Physics

Recently, Hammond and Sheffield [Probab. Theory Related Fields 157 (2013) 691–719] introduced a model of correlated one-dimensional random walks that scale to fractional Brownian motions with long-range dependence. In this paper, we consider a natural generalization of this model to dimension $d\geq2$. We define a $\mathbb{Z}^{d}$-indexed random field with dependence relations governed by an underlying random graph with vertices $\mathbb{Z}^{d}$, and we study the scaling limits of the partial sums of the random field over rectangular sets. An interesting phenomenon appears: depending on how fast the rectangular sets increase along different directions, different random fields arise in the limit. In particular, there is a critical regime where the limit random field is operator-scaling and inherits the full dependence structure of the discrete model, whereas in other regimes the limit random fields have at least one direction that has either invariant or independent increments, no longer reflecting the dependence structure in the discrete model. The limit random fields form a general class of operator-scaling Gaussian random fields. Their increments and path properties are investigated.

Based on the analysis of stratification structure on random normed modules, we first present the notion of random smoothness in random normed modules. Then, we establish the relations between random smoothness and random strict convexity. Finally, a type of Gâteaux differentiability is defined for random norms, and its relation to random smoothness is given. The results are helpful in the further study of geometry of random normed modules.

This paper studies linear set-dynamics driven by random convex compact sets (RCCSs), where the parameter matrix evolves according to an underlying Markovian random process taking values in a finite set. We derive dynamics of the expectations of the associated reach sets. We establish that such expectations evolve according to coupled deterministic set-dynamics. We provide sufficient conditions for the convergence of the reach sets expectations. We also give conditions under which the reach sets remain asymptotically bounded with probability one. As an illustrative example, we apply our results to evaluate the expectations of the reach sets associated to the position of a quadrotor.

Let $X$ be a locally compact Abelian group, $Y$ be its character group. Following A. Kagan and G. Sz\'ekely we introduce a notion of $Q$-independence for random variables with values in $X$. We prove group analogues of the Cram\'er, Kac-Bernstein, Skitovich-Darmois and Heyde theorems for $Q$-independent random variables with values in $X$. The proofs of these theorems are reduced to solving some functional equations on the group $Y$.

In this chapter we study how to generate a well-founded set of “size” n, uniformly at random. To wit, we will see algorithms that, given n, produce a well-founded set of size n at random, so that each set of size n has equal probability to occur. Procedures of this kind can be of use for testing the correctness of algorithm implementations, or for testing conjectures about data that is reasonably represented by well-founded sets.This chapter also serves the purpose of describing three general methods of generating combinatorial objects uniformly at random. The first two of these are based on a so-called combinatorial decomposition of the objects. The third random generation method is based on a Markov chain. Such technique has the advantage that it requires less insight into the structure of the objects to be generated. It suffers a major drawback, though, namely that it is often hard to establish for how long the Markov-chain stochastic process must be iterated in order to ensure that objects are generated with uniform probability distribution.

We derive general existence theorems for random pullback exponential attractors and deduce explicit bounds for their fractal dimension. The results are formulated for asymptotically compact random dynamical systems in Banach spaces. Fondo Europeo de Desarrollo Regional Ministerio de Economía y Competitividad Junta de Andalucía

[EN] Generalized polynomial chaos (gPC) is a spectral technique in random space to represent random variables and stochastic processes in terms of orthogonal polynomials of the Askey scheme. One of its most fruitful applications consists of solving random differential equations. With gPC, stochastic solutions are expressed as orthogonal polynomials of the input random parameters. Different types of orthogonal polynomials can be chosen to achieve better convergence. This choice is dictated by the key correspondence between the weight function associated to orthogonal polynomials in the Askey scheme and the probability density functions of standard random variables. Otherwise, adaptive gPC constitutes a complementary spectral method to deal with arbitrary random variables in random differential equations. In its original formulation, adaptive gPC requires that both the unknowns and input random parameters enter polynomially in random differential equations. Regarding the inputs, if they appear as non-polynomial mappings of themselves, polynomial approximations are required and, as a consequence, loss of accuracy will be carried out in computations. In this paper an extended version of adaptive gPC is developed to circumvent these limitations of adaptive gPC by taking advantage of the random variable transformation method. A number of illustrative examples show the superiority of the extended adaptive gPC for solving nonlinear random differential equations. In addition, for the sake of completeness, in all examples randomness is tackled by nonlinear expressions., This work has been partially supported by the Ministerio de Economia y Competitividad grants MTM2013-41765-P.

Network analysis has benefited greatly from published data of social networks. However, the privacy of users may be compromised even if the data are released after applying anonymization techniques. To measure the resistance against privacy attacks in an anonymous network, Trujillo-Rasua R. et al. introduce the concepts of k-antiresolving set and k-metric antidimension [1]. In this paper, we prove that the problem of k-metric antidimension is NP-hard. We also study the size of k-antiresolving sets in random graphs. Specifically, we establish three bounds on the size of k-antiresolving sets in Erdős-Renyi random graphs.

[EN] This paper extends both the deterministic fractional Riemann¿Liouville integral and the Caputo fractional derivative to the random framework using the mean square random calculus. Characterizations and sufficient conditions to guarantee the existence of both fractional random operators are given. Assuming mild conditions on the random input parameters (initial condition, forcing term and diffusion coefficient), the solution of the general random fractional linear differential equation, whose fractional order of the derivative is ¿ ¿ [0, 1], is constructed. The approach is based on a mean square chain rule, recently established, together with the random Fröbenius method. Closed formulae to construct reliable approximations for the mean and the covariance of the solution stochastic process are also given. Several examples illustrating the theoretical results are included., This work has been partially supported by the Ministerio de Economia y Competitividad grant MTM2013-41765-P. The co-author Prof. L. Villafuerte acknowledges the support by Mexican Conacyt.

As the name suggests, a random set is an object with values being sets, so that the corresponding record space is the space of subsets of a given carrier space. At this stage, a mere definition of a general random element like a random set presents little difficulty as soon as a σ-algebra on the record space is specified.

[EN] This paper deals with the explicit determination of the first probability density function of the solution stochastic process to random autonomous first-order linear systems of difference equations under very general hypotheses. This finding is applied to extend the classical stability classification of the zero-equilibrium point based on phase portrait to the random scenario. An example illustrates the potentiality of the theoretical results established and their connection with their deterministic counterpart., This work has been partially supported by the Ministerio de Economia y Competitividad grant MTM2013-41765-P. Ana Navarro Quiles acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigation y Desarrollo (PAID), Universitat Politecnica de Valencia.

We describe a list of open problems in random matrix theory and the theory of integrable systems that was presented at the conference Asymptotics in Integrable Systems, Random Matrices and Random Processes and Universality, Centre de Recherches Mathematiques, Montreal, June 7-11, 2015. We also describe progress that has been made on problems in an earlier list presented by the author on the occasion of his 60th birthday in 2005 (see [Deift P., Contemp. Math., Vol. 458, Amer. Math. Soc., Providence, RI, 2008, 419-430, arXiv:0712.0849])., for Part I see arXiv:0712.0849

This paper discusses random utility representations of the Luce model (Luce, Individual choice behavior: a theoretical analysis, 1959). Earlier works, such as McFadden (Frontier in econometrics, 1973), Yellott (J Math Psychol 15:109–144, 1977), and Strauss (J Math Psychol 20:35–52, 1979) have discussed random utility representations under the assumption that utilities are additively (or multiplicatively) separable in a deterministic and a random part. Under various conditions, they have established that a separable and independent random utility representation exists if and only if the random terms are type III (type I) extreme value distributed. This paper analyzes independent random utility representations without the separability condition and with an infinite universal set of alternatives. Under these assumptions, it turns out that the most general random utility representation of the Luce model is a utility function that is an arbitrary strictly increasing transformation of a separable utility function (additive or multiplicative) with extreme value distributed random terms.

We study the asymptotic behavior of solutions to a class of non-autonomous stochastic lattice systems driven by multiplicative white noise. We prove the existence and uniqueness of tempered random attractors in a weighted space containing all bounded sequences, and establish the upper semicontinuity of random attractors as the intensity of noise approaches zero. We also construct maximal and minimal tempered random complete solutions which bound the attractors from above and below, respectively. When deterministic external forcing terms are periodic in time, we show the random attractors are pathwise periodic. In addition, we exhibit a non-autonomous stochastic lattice system which possesses an infinite-dimensional tempered random attractor.

In this paper, we investigate the Koopman and the Perron-Frobenius operators of random dynamical systems in general setting. Moreover, we give some necessary and sufficient conditions for the existence of invariant densities for the associated Perron-Frobenius operators.

A sufficient condition for a family of random attractors to be upper semi-continuous and regular is obtained when the state space is a p -times integrable space with p > 2 . It is shown that both upper semi-continuity and regularity under the topology of p -norms do not rely on continuity, convergence and absorption of the associated random dynamical systems in L p since the corresponding L 2 -properties can offer all. As an application of the abstract result, it is shown that the family of random attractors for the stochastic reaction–diffusion equations on a unbounded domain is upper semi-continuous and regular at any point under the topology of p -norms.

Let {Xnk,n≥1,k≥1} be a ρ--mixing random sequence, by using the moment inequality and truncation method, we studied the weak convergence of random sums for ρ--mixing random sequence, which extended and improved some results in related literature.

International audience; Generalizations of various random tessellation models generated by Poisson point processes are introduced, and their functional probability P(K) is given. They are obtained from Boolean random function models, and alternatively from a geodesic distance, providing a generic way of simulation of a wide range of random tessellations, as illustrated in the paper.

We consider random fractal sets with random recursive con- structions in which the contracting vectors have different distributions at different stages. We prove that the random fractal associated with such construction has a constant packing dimension almost surely and give an explicit formula to determine it.

An approximation scheme for estimating a fixed, unknown fuzzy set from random samples taken from the nested random set defined by its α-level sets is presented. Its strong consistency is studied, giving rates of convergence in four metrics. A simulation study suggests that the behaviour for moderately small samples is coherent with the theoretical rate of convergence valid for large samples.

The aim of this paper is to study weak and strong convergence of an implicit random iterative process with errors to a common random fixed point of two finite families of asymptotically nonexpansive random mappings in a uniformly convex separable Banach space.

We study the regularity of random attractors for a class of stochastic degenerate parabolic equations with the leading term involving a diffusion variable σ which many be non-smooth or unbounded. Without any restrictions on the upper growth order p of the nonlinearity, except that p ⩾ 2 , we show that the associated random dynamical system admits a unique compact random attractor in the space D 0 1 , 2 ( D N , σ ) ∩ L ϖ ( D N ) for any ϖ ∈ [ 2 , 2 p - 2 ] , where D N is an arbitrary (bounded or unbounded) domain in R N , N ⩾ 2 .

Jensen's inequality is extended to metric spaces endowed with a convex combination operation. Applications include a dominated convergence theorem for both random elements and random sets, a monotone convergence theorem for random sets, and other results on set-valued expectations in metric spaces and on random probability measures.

Let L E 2 $L^2_E$ be a Banach space of all ℙ-square integrable random variables defined on some probability space ( Ω , 𝒜 , ℙ ) $(\Omega ,\mathcal {A},\mathbb {P})$ with values in the complex separable Banach space E. For an L E 2 $L_{E}^{2}$ -valued stationary continuous random function 𝔛 = ( X g ) g ∈ G $\mathfrak {X}=(X_{g})_{g\in G}$ on the locally compact abelian group G, we provide a condition of existence of a random measure Z taking values in L E 2 $L^{2}_{E}$ such that ( X g ) g ∈ G $(X_{g})_{g\in G}$ has a spectral representation as a Fourier transform of Z.

We obtain a new generalization of Chebyshev’s inequality for random vectors. Then we extend this result to random elements taking values in a separable Hilbert space.

We study an Achlioptas-process version of the random k-SAT process: a bounded number of k-clauses are drawn uniformly at random at each step, and exactly one added to the growing formula according to a particular rule. We prove the existence of a rule that shifts the satisfiability threshold. This extends a well-studied area of probabilistic combinatorics Achlioptas processes to random CSP's. In particular, while a rule to delay the 2-SAT threshold was known previously, this is the first proof of a rule to shift the threshold of k-SAT for ki¾?3. We then propose a gap decision problem based upon this semi-random model. The aim of the problem is to investigate the hardness of the random k-SAT decision problem, as opposed to the problem of finding an assignment or certificate of unsatisfiability. Finally, we discuss connections to the study of Achlioptas random graph processes. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 163-173, 2015

The purpose of this paper is to examine the notion of completely random operators and to present some results on the existence of random coincidence points of completely random operators. Some applications to random fixed point theorems and random equations are given.

We establish stability of random absolutely continuous invariant measures (acims) for cocycles of random Lasota-Yorke maps under a variety of perturbations. Our family of random maps need not be close to a fixed map; thus, our results can handle very general driving mechanisms. We consider (i) perturbations via convolutions, (ii) perturbations arising from finite-rank transfer operator approximation schemes and (iii) static perturbations, perturbing to a nearby cocycle of Lasota-Yorke maps. The former two results provide a rigorous framework for the numerical approximation of random acims using a Fourier-based approach and Ulam's method, respectively; we also demonstrate the efficacy of these schemes.

A probability measure Pn on the symmetric group S n is said to be record-dependent if P n ( σ ) depends only on the set of records of a permutation σ ∈ S n . A sequence P = ( P n ) n ∈ ℕ of consistent record-dependent measures determines a random order on ℕ. In this paper we describe the extreme elements of the convex set of such P. This problem turns out to be related to the study of asymptotic behavior of permutation-valued growth processes, to random extensions of partial orders, and to the measures on the Young-Fibonacci lattice. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 2014 © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 46,688–706, 2015