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This paper is an extension of known results of Pesin’s entropy formula and SRB measures for random compositions of infinite-dimensional mappings to the continuous-time setting of stochastic flows. Consider a stochastic flow ϕ on a separable infinite dimensional Hilbert space preserving a probability measure μ, which is supported on a random compact set K. We show that if ϕ is C2 (on K) and satisfies some mild integrable conditions of the differentials, then Pesin’s entropy formula holds if μ has absolutely continuous conditional measures along the unstable manifolds. The converse is also true under an additional condition on K when the system has no zero Lyapunov exponent.

In this paper, we study several definitions of topological entropy (of noncompact or noninvariant bundles) for random transformations on a compact metric space. Furthermore their relationships are established.

In this paper, generalized random linear mappings are viewed as random field indexed by random parameters. This point of view leads to questions about sample paths properties. We obtain various necessary and sufficient conditions ensuring the existence of nice sample paths of a generalized random linear mapping. We also give illustrative examples and applications.

A quasi-perfect map is a continuous, closed function such that the preimage of every point is countably compact. An ambitious old problem due to van Douwen [1] is whether every first countable regular space of cardinality ≤ c is a quasi-perfect image of a locally compact space. Here we construct locally compact, normal, quasi-perfect preimages for all stationary, co-stationary subsets of ω 1 . These subsets are ω 1 -compact but not σ-countably compact. Quasi-perfect preimages preserve these properties, and the preimages constructed here are all of cardinality b . This provides a new upper bound for the following problem: What is the least cardinality of a ZFC example of a locally compact, ω 1 -compact space that is not σ-countably compact?

In this paper, we suggest the modified random S-iterative process and prove the common random fixed point theorems of a finite family of random uniformly quasi-Lipschitzian operators in a generalized convex metric space. Our results improves and extends various results in the literature.

The uncertainty related to the spatially varying random property requires to be described as a random field, which in definition consists of an infinite number of random variables that are attached to each point of a continuous domain of the random field. The utility of Karhunen–Loeve expansion allows to represent the random field as the summation of a series of deterministic functions and random variables. By this way, the random field is represented by using a finite number of random variables. However, the K–L expansion depends crucially upon the eigen-solutions of the Fredholm integral equation of the second kind. Therefore, the interest of practical random field simulations is mainly concerned with containing as few random variables as possible, but needs to guarantee small approximation errors for the reproduced results in the meantime. To this end, the Chebyshev polynomial is introduced in the paper for the solution of the integral eigenvalue problem. As an effective class of basis functions for the Galerkin projection, the convergence rate of the Chebyshev polynomial-based Galerkin method is assessed by means of the global variance and covariance errors. And its accuracy is examined by reproducing several stationary and non-stationary, Gaussian, and non-Gaussian random fields. Together with the practical simulations of the concrete compressive strength field, numerical results show that the Chebyshev polynomial-based Galerkin scheme has engineering applications.

Deterministic chaos is investigated for random dynamical systems in dimension one. Some well-known as well as new Li-Yorke-type theorems are randomized. Deterministic chaos exhibited by random dynamics is therefore called random chaos for brevity. Chaotic random dynamics are also studied for multivalued maps.

The long-time behavior of solutions (more precisely, the existence of random pullback attractors) for an integro-differential parabolic equation of diffusion type with memory terms, more particularly with terms containing both finite and infinite delays, as well as some kind of randomness, is analyzed in this paper. We impose general assumptions not ensuring uniqueness of solutions, which implies that the theory of multivalued dynamical system has to be used. Furthermore, the emphasis is put on the existence of random pullback attractors by exploiting the techniques of the theory of multivalued nonautonomous/random dynamical systems.

We introduce a new class of non-Gaussian vector random fields in space and/or time, which are termed binomial-$\chi^2$ vector random fields and include $\chi^2$ vector random fields as special case...

In this paper, we establish a unique common random fixed point theorem in cone random metric spaces for four weakly compatible mappings by using an implicit relation. Some corollaries of this theorem for two and three random weakly compatible mappings are obtained. Some examples are given to support our generalization. Our results presented in this paper extend and improve several recent results in the setting of cone random metric spaces.

In this paper, we prove the existence of common random fixed point for two random operators under general quasi contraction condition in a complete p-normed space X (with whose dual separates the point of X). Also, the well-posedness problem of random fi...

In this note, we prove the conjecture in the paper [Shang Y. \emph{A central limit theorem for randomly indexed m-dependent random variables.} Filomat 26.4 (2012): 713-717] about the sum of a random number $N_n$ of $m$-dependent random variables. The random number $N_n$ is supposed to converge in probability toward a positive random variable.

We propose a model of a one-dimensional random walk in dynamic random environment that interpolates between two classical settings: (I) the random environment is sampled at time zero only; (II) the random environment is resampled at every unit of time. In our model the random environment is resampled along an increasing sequence of deterministic times. We consider the annealed version of the model, and look at three growth regimes for the resampling times: (R1) linear; (R2) polynomial; (R3) exponential. We prove weak laws of large numbers and central limit theorems. We list some open problems and conjecture the presence of a crossover for the scaling behaviour in regimes (R2) and (R3).

We define two non-linear operations with random (not necessarily closed) sets in Banach space: the conditional core and the conditional convex hull. While the first is sublinear, the second one is superlinear (in the reverse set inclusion ordering). Furthermore, we introduce the generalised conditional expectation of random closed sets and show that it is sandwiched between the conditional core and the conditional convex hull. The results rely on measurability properties of not necessarily closed random sets considered from the point of view of the families of their selections. Furthermore, we develop analytical tools suitable to handle random convex (not necessarily compact) sets in Banach spaces; these tools are based on considering support functions as functions of random arguments. The paper is motivated by applications to assessing multivariate risks in mathematical finance., 31 page, this work is a substantial extension of a part from arXiv:1605.07884

We introduce the notion of a random symmetry. It consists of taking the action given by a deterministic flow that maintains the solutions of a given differential equation invariant and replacing it with a stochastic flow. This generates a random action, which we call a random symmetry.

The existence of a compact random attractor for the stochastic complex Ginzburg–Landau equation with multiplicative noise has been investigated on unbounded domain. The solutions are considered in suitable spaces with weights. According to crucial properties of Ornstein–Uhlenbeck process, using the tail-estimates method, the key uniform a priori estimates for the tail of solutions have been obtained, which give the asymptotic compactness of random attractors. Then the existence of a compact random attractor for the corresponding dynamical system is proved in suitable spaces with weights.

In this paper, we prove a unique common ran- dom fixed p oint t heorem i n t he f ramework o f c one random metric spaces for four weakly random compatible mappings under strict contractive condition. Some corollaries of this theorem for three and two weakly random compatible mappings and for one random mapping are derived. Two examples to justify our theorem are given. Our results extend some previous work related to cone random metric spaces from the current existing literature.

Finite sample properties of random covariance-type matrices have been the subject of much research. In this paper we focus on the “lower tail” of such a matrix, and prove that it is sub-Gaussian under a simple fourth moment assumption on the one-dimensional marginals of the random vectors. A similar result holds for more general sums of random positive semidefinite matrices, and our (relatively simple) proof uses a variant of the so-called PAC-Bayesian method for bounding empirical processes. Using this bound, we obtain a nearly optimal finite-sample result for the ordinary least squares estimator under random design.

In this paper, we consider the asymptotic behavior of solutions for stochastic non-autonomous FitzHugh–Nagumo system with multiplicative white noise. First we prove the existence of random attractor of the random dynamical system generated by the solutions of considered system. Then we present some conditions for estimating an upper bound of the fractal dimension of a random invariant set of a random dynamical system on a separable Banach space and apply these conditions to prove the finiteness of fractal dimension of random attractor.

In this paper we want to show the existence of a form of asymptotic stability of random dynamical systems in the sense of L. Arnold using arguments analogous to those presented by T. Szarek in [6], that is showing it using conditions generalizing the notion of tightness of measures. In order to do that we use tightness theory for random measures as developed by H. Crauel in [2].

In this article, a convolution of second-order random measures on a locally compact abelian groups is considered. A random measure is an analog of a stochastic process. With a newly defined (random) convolution, we construct a ring and a normed algebra of second-order random measures.

In this paper, we establish some common random fixed point theorems for contractive type conditions in the setting of cone random metric spaces. Our results unify, extend and generalize many known results from the current existing literature.

When developing statistical models, it is of fundamental importance to decide if the various components are independent of one another, preferably using a formal statistical test. Non-parametric versions of such tests are particularly useful, as they do not require extensive a priori knowledge about the underlying models. In this paper, we develop such tests for random marked closed sets, which have many applications in spatial statistics. More precisely, we investigate two approaches to testing if the marks are independent of the closed set. Both approaches are based on second-order characteristics of random marked closed sets. The first approach uses a global rank envelope test based on the mark-weighted K -function. The second approach uses an asymptotic test developed for marked point processes. We carry out extensive simulation studies to assess the performance of these tests, demonstrating that the global rank envelope test is a better choice. Finally, we apply this test to two real world data sets.

The aim of this paper is to consider reversible random walk in a random environment in one dimension and prove the Einstein relation for this model. It says that the derivative at 0 of the effective velocity under an additional local drift equals the diffusivity of the model without drift (Theorem 1.2). Our method here is very simple: we solve the Poisson equation (Pω−I)g=f and then use the pointwise ergodic theorem in Wiener (1939) [10] to treat the limit of the solutions to obtain the desired result. There are analogous results for Markov processes with discrete space and for diffusions in random environment.

A significant amount of problems and applications in stochastic mechanics and engineering involve multi-dimensional random functions. The probabilistic analysis of these problems is usually computationally very expensive if a brute-force Monte Carlo method is used. Thus, a technique for the optimal selection of a moderate number of samples effectively representing the entire space of sample realizations is of paramount importance. Functional Quantization is a novel technique that has been proven to provide optimal approximations of random functions using a predetermined number of representative samples. The methodology is very easy to implement and it has been shown to work effectively for stationary and non-stationary one-dimensional random functions. This paper discusses the application of the Functional Quantization approach to the domain of multi-dimensional random functions and the applicability is demonstrated for the case of a 2D non-Gaussian field and a two-dimensional panel with uncertain Young modulus under plane stress.

In this paper, we first give some conditions for bounding the fractal dimension of a random invariant set for a non-autonomous random dynamical system on a separable Banach space. Then we apply these conditions to prove the finiteness of fractal dimension of the random attractors for stochastic reaction-diffusion equations with multiplicative white noise and additive white noise.

In this paper, we first establish a criteria for estimating the upper bound of fractal dimension of a random compact invariant set of a non-autonomous random dynamical system on a separable Banach space and give some special conditions which can be easily verified to some stochastic partial differential equations for bounding the fractal dimension of random attractors. Then we apply these conditions to obtain an upper bound of fractal dimension of random attractor for stochastic damped wave equation with additive white noise.

RANDOM LINEAR OPERATORS ONPROBABILISTIC UNITARY SPACES TranXuan Quy, DangHung Thang,and NguyenThinh Abstract. In this paper, we are concerned with abstract random linearoperators on probabilistic unitary spaces which are a generalization ofgeneralized random linear operators on a Hilbert space defined in [25].The representation theorem for abstract random bounded linear operatorsand some results on the adjoint of abstract random linear operators aregiven. 1. IntroductionLet (Ω,F,P) be a complete probability space and X,Y be Banach spaces.A mapping f : Ω × X → Y is said to be a random operator (or a randommapping) defined on X with values in Y if for each x ∈ X, the mappingω → f(ω,x) is a Y-valued random variable. Equivalently, a random operatordefined on X with values in Y is a mapping from X into the space L Y0 (Ω)of all Y -valued random variables. A random operator f : X → L Y0 (Ω) issaid to be a random linear operator if f is linear. The interest in randomoperators has been arouse not only for its own right as a random generalizationof usual deterministic operators but also for their widespread applications inother areas. Research in theory of random operators has been carried out inmany directions including random linear operators which provide a frameworkof stochastic integral, infinite random matrix (see e.g. [1], [2], [15]-[20], [23]-[26]), random fixed points of random operators and random operator equations,(e.g [3]-[14], [18], [21], [22] and references therein). As an extension of randomlinear operators, generalized random linear operators on a separable Hilbertspace were introduced and investigated in [25].In this paper, generalized random linear operators on a separable Hilbertspace are extended to abstract random linear operators on probabilistic unitaryspaces. Section 2 presents the definitions and some properties of probabilistic

This paper addresses the problem of simulating multivariate random fields with stationary Gaussian increments in a d-dimensional Euclidean space. To this end, one considers a spectral turning-bands algorithm, in which the simulated field is a mixture of basic random fields made of weighted cosine waves associated with random frequencies and random phases. The weights depend on the spectral density of the direct and cross variogram matrices of the desired random field for the specified frequencies. The algorithm is applied to synthetic examples corresponding to different spatial correlation models. The properties of these models and of the algorithm are discussed, highlighting its computational efficiency, accuracy and versatility.

Gaps (or spacings) between consecutive eigenvalues are a central topic in random matrix theory. The goal of this paper is to study the tail distribution of these gaps in various random matrix models. We give the first repulsion bound for random matrices with discrete entries and the first super-polynomial bound on the probability that a random graph has simple spectrum, along with several applications., Several typos corrected, new references and a new application added (40 pages)

In this paper, we shall present the strong laws of large numbers for fuzzy set-valued random variables in the sense of d∞H . The results are based on the result of single-valued random variables obtained by Taylor [1] and set-valued random variables obtained by Li Guan [2].

Random compact operators are useful to study random differentiation and random integral equations. In this paper, we define the random norm of R-bounded operators and study random norms of differentiation operators and integral operators. The definition of random norm of R-bounded operators led us to study the random operator theory.

In this paper we investigate the long term behavior of a stochastic lattice dynamical system with a diffusive nearest neighbor interaction, a dissipative nonlinear reaction term, and a different multiplicative white noise at each node. We prove that this stochastic lattice equation generates a random dynamical system that possesses a global random attractor. In particular, we first establish an existence theorem for weak solutions to general random evolution equations, which is later applied to the specific stochastic lattice system to show that it has weak solutions and the solutions generate a random dynamical system. We then prove the existence of a random attractor of the underlying random dynamical system by constructing a random compact absorbing set and using an embedding theorem. The major novelty of this work is that we consider a different multiplicative white noise term at each different node, which significantly improves the previous results in the literature where the same multiplicative noise was considered at all the nodes. As a consequence, the techniques used in the existing literature are not applicable here and a new methodology has to be developed to study such systems.

We consider integrals of continuous random functions over random measures induced by level sets of a Gaussian random field. We show that under some conditions on the generating Gaussian field, such integrals form a continuous random process indexed by level sets. If the integrand random field possesses certain weak dependence conditions, a functional central limit theorem for those processes is proved.

We consider a random sphere covering model made of random balls with interacting random radii of the product form R(r,ω) = rG(ω), based on a Poisson random measure ω(dy,dr) on Rd × R+. We provide sufficient conditions under which the corresponding random ball counting processes are well-defined, and we study the fractional behavior of the associated random fields. The main results rely on moment formulas for Poisson stochastic integrals with random integrands. Accepted version

In this paper, we consider the existence of a pullback attractor for the random dynamical system generated by stochastic two-compartment Gray-Scott equation for a multiplicative noise with the homogeneous Neumann boundary condition on a bounded domain of space dimension n ≤ 3. We first show that the stochastic Gray-Scott equation generates a random dynamical system by transforming this stochastic equation into a random one. We also show that the existence of a random attractor for the stochastic equation follows from the conjugation relation between systems. Then, we prove pullback asymptotical compactness of solutions through the uniform estimate on the solutions. Finally, we obtain the existence of a pullback attractor.

Models are developed for random functions Q ( x , t ) of space x ∈ D and time t ∈ [ 0 , τ ] from samples of these functions and any other information when available. Most of the models in the paper can be viewed as extensions of Karhunen–Loeve (KL) representations for random fields. Their samples are linear forms of basis functions with random coefficients which are extracted from samples of Q ( x , t ) by singular value decomposition. The coefficients of these forms are stochastic processes rather than random variables. The proposed models can be used to generate large sets of samples whose statistics are similar to those of target random functions. Theoretical arguments and numerical examples are presented to establish properties of the proposed models, assess their accuracy, and illustrate their implementation.

Branko Ristic has given us a superb new book that combines two hot topics (particle filters and random sets) into a delicious feast of algorithms, Monte Carlo simulations, important applications, and mathematics that is new for normal engineers, all served up with clarity and thoroughness. The seminal research on random sets applied to tracking and data fusion problems is due to Ron Mahler [1]. It is clearly a good idea to apply random sets to such problems; in fact, it is hard to avoid using them either implicitly or explicitly. For example, the set of hypotheses for the well-known multiple hypothesis tracking (MHT) algorithms are random sets. The set of hypotheses for MHT is obviously random, because it depends on the measurements (which are random owing to noise errors and random target states). This is exactly the viewpoint adopted in [5], in which one is forced to compute equivalence classes of random sets in order to obtain a set of hypotheses that is mutually exclusive and collectively exhaustive, as required for any rational probability model of the situation (Bayesian or otherwise). It turns out that “fuzzy sets” correspond to equivalence classes of random sets, as explained by Goodman [6]. The set of particles in particle filters is also random, because it is generated from random numbers, but almost no papers explicitly use non-trivial random set theory for particle filter design. In fact, there are very few papers that explicitly talk about random sets and particle filters together, with the conspicuous exception of [20]. Nevertheless, random sets are a mature and well documented mathematical tool, as shown in [1], [2], [3], and [4]. You should not be afraid of random set mathematics; Ron Mahler has written a series of highly accessible tutorials on using random finite sets which avoid the pathologies and complexity of random unfettered sets (e.g., [26] and [28]).

In this paper, we consider the asymptotic behaviour of solutions to second-order non-autonomous stochastic lattice equations with dispersive term and additive white noises in the space of infinite sequences. We first transfer the stochastic lattice equations into random lattice equations, and prove the existence and uniqueness of solutions that generate a random dynamical system. Second, we prove the existence of a tempered random absorbing set and a random attractor for the system. Finally, we establish the upper semi-continuity of the random attractors as the coefficient of the white noise term tends to zero.

We prove the existence of families of random invariant manifolds for singularly perturbed systems of ordinary differential equations with sufficiently small real noise. We use these invariant manifolds to prove a random version of the inclination theorem or exchange lemma.