Random Compact Sets Research Articles

Binary robustness of random attractors for 2D-Ginzburg–Landau equations with Wong–Zakai noise

Consider a non-autonomous 2D-Ginzburg–Landau equation driven by Wong–Zakai noise or white noise, respectively, we first show the existence of pullback random attractors, which are random compact attracting sets indexed by two parameters: the size of Wong–Zakai noise and the current time. We then establish the robustness of the attractors when both parameters are simultaneously convergent. An essential difficulty arises from the possible loss of the convergence of solutions and only part convergence of solutions is available, which is a new phenomenon for 2D-GL equation distinguishing with the 1D case. So, by using part joint-convergence, regularity, eventual local-compactness and recurrence, we establish a binary robustness theorem of pullback random attractors and apply it to the weakly dissipative stochastic equation.

Residually continuous pullback attractors for stochastic discrete plate equations

For a nonautonomous stochastic discrete plate equation driven by multiplicative noise, we prove the unique existence of a pullback random attractor, which is a family of pullback‐attracting random compact sets parameterized by time and samples. We then establish the residual dense continuity of the pullback random attractor on the time‐sample plane with respect to the Hausdorff metric. Even without the Cantor continuum hypothesis, we show that the set of all points of continuity of the pullback random attractor is an uncountable set with the continuity cardinality. These results explain both nonexplosive and nonimplosive phenomena for the random plate lattice system.

Existence and upper semicontinuity of random attractors for the 2D stochastic convective Brinkman–Forchheimer equations in bounded domains

In this work, we discuss the large time behaviour of the solutions of two-dimensional stochastic convective Brinkman–Forchheimer (SCBF) equations on bounded domains. Under the functional setting , where and are appropriate separable Hilbert spaces and the fact that the embedding is compact, we establish the existence of random attractors in for the stochastic flow generated by 2D SCBF equations perturbed by additive noise. We prove the upper semicontinuity of the random attractors for 2D SCBF equations in , when the coefficient of random term approaches zero. Moreover, we obtain the existence of random attractors in a more regular space , using the pullback flattening property. The existence of random attractors ensures the existence of invariant compact random set and hence we show the existence of an invariant measure for 2D SCBF equations. Finally, we also comment on the uniqueness of invariant measures.

Convergence theorems for random elements in convex combination spaces

A Vitali convergence theorem is proved for subspaces of an abstract convex combination space which admits a complete separable metric. The convergence may be in that metric or, more generally, in a quasimetric satisfying weaker properties. Versions for convergence in probability and in distribution are given. As applications, we show that some dominated convergence theorems in the literature of fuzzy random variables and random compact sets can be recovered or improved, and we derive new convergence theorems in another space of sets and in a space of probability distributions.

Finite-Dimensionality of Tempered Random Uniform Attractors

Finite-dimensional attractors play an important role in finite-dimensional reduction of PDEs in mathematical modelization and numerical simulations. For non-autonomous random dynamical systems, Cui and Langa (J Differ Equ, 263:1225–1268, 2017) developed a random uniform attractor as a minimal compact random set which provides a certain description of the forward dynamics of the underlying system by forward attraction in probability. In this paper, we study the conditions that ensure a random uniform attractor to have finite fractal dimension. Two main criteria are given, one by a smoothing property and the other by a squeezing property of the system, and neither of the two implies the other. The upper bound of the fractal dimension consists of two parts: the fractal dimension of the symbol space plus a number arising from the smoothing/squeezing property. As an illustrative application, the random uniform attractor of a stochastic reaction–diffusion equation with scalar additive noise is studied, for which the finite-dimensionality in L^2 is established by the squeezing approach and that in H_0^1 by the smoothing framework. In addition, a random absorbing set that absorbs itself after a deterministic period of time is also constructed.

A new look at random projections of the cube and general product measures

A consequence of the celebrated Dvoretzky–Milman theorem is a strong law of large numbers for d-dimensional random projections of the n-dimensional cube. It shows that, with respect to the Hausdorff distance, a uniform random projection of the cube [−1/n,+1/n]n onto Rd converges almost surely to a centered d-dimensional Euclidean ball of radius 2/π, as n→∞. We start by providing an alternative proof of this strong law via the Artstein–Vitale law of large numbers for random compact sets. Then, for every point inside the ball of radius 2/π, we determine the asymptotic number of vertices and the volume of the part of the cube projected ‘close’ to this point. More generally, we study large deviations for random projections of arbitrary product measures. Let ν⊗n be the n-fold product measure of a Borel probability measure ν on R, and let I be uniformly distributed on the Stiefel manifold of orthogonal d-frames in Rn. It is shown that the sequence of random measures ν⊗n∘(n−1/2I∗)−1, n∈N, satisfies a large deviation principle with probability 1. The rate function is explicitly identified in terms of the moment generating function of ν. At the heart of the proofs lies a transition trick which allows to replace the uniform projection by the Gaussian one. A number of concrete examples are discussed as well, including the uniform distributions on the cube [−1,1]n and the discrete cube {−1,1}n as special cases.

Random attractors via pathwise mild solutions for stochastic parabolic evolution equations

We investigate the longtime behavior of stochastic partial differential equations (SPDEs) with differential operators that depend on time and the underlying probability space. In particular, we consider stochastic parabolic evolution problems in Banach spaces with additive noise and prove the existence of random exponential attractors. These are compact random sets of finite fractal dimension that contain the global random attractor and are attracting at an exponential rate. In order to apply the framework of random dynamical systems, we use the concept of pathwise mild solutions.

Double stabilities of pullback random attractors for stochastic delayed p‐Laplacian equations

We provide a method to study the double stabilities of a pullback random attractor (PRA) generated from a stochastic partial differential equation (PDE) with delays, such a PRA is actually a family of compact random sets Aϱ(t,·), where t is the current time and ϱ is the memory time. We study its longtime stability, which means the attractor semiconverges to a compact set as the current time tends to minus infinity, and also its zero‐memory stability, which means the delayed attractor semiconverges to the nondelayed attractor as the memory time tends to zero. The stochastic nonautonomous p‐Laplacian equation with variable delays on an unbounded domain will be applied to illustrate the method and some suitable assumptions about the nonlinearity and time‐dependent delayed forces can ensure existence, backward compactness, and double stabilities of a PRA.

Long time behavior for the stochastic parabolic-wave systems with delay on infinite lattice

In this paper, we consider the existence of random attractors for the stochastic parabolic-wave systems with delay on infinite lattice. Under suitable dissipative conditions, it shows that such a system has a random attractor which is a random compact invariant set.

A Remark on Stochastic Flows in a Hilbert Space

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.

Oriented distance point of view on random sets

Motivated by free boundary problems under uncertainties, we consider the oriented distance function as a way to define the expectation for a random compact or open set. In order to provide a law of large numbers and a central limit theorem for this notion of expectation, we also address the question of the convergence of the level sets offnto the level sets offwhen (fn) is a sequence of functions uniformly converging tof. We provide error estimates in term of Hausdorff convergence. We illustrate our results on a free boundary problem.

Testing Equality of Distributions of Random Convex Compact Sets via Theory of $\mathfrak {N}$-Distances

This paper concerns a method of testing the equality of distributions of random convex compact sets. The main theoretical result involves a construction of a metric on the space of distributions of random convex compact sets. We obtain it by using the theory of \(\mathfrak {N}\)-distances and the redefined characteristic function of random convex compact set. We propose an approximation of the metric through its finite-dimensional counterparts. This result leads to a new statistical test for testing the equality of distributions of two random convex compact sets. Consequently, we show a heuristic approach how to determine whether two realisations of random sets that can be approximated by a union of identically distributed random convex compact sets come from the same underlying process using the constructed test. Each procedure is justified by an extensive simulation study and the heuristic method for comparing random sets using their convex compact counterparts is moreover applied to real data concerning histological images of two different types of mammary tissue.

Approximation by random complex polynomials and random rational functions

We investigate random compact sets with random functions defined thereon, such as polynomials, rational functions, the pluricomplex Green function and the Siciak extremal function. One surprising consequence of our study is that randomness can be used to `improve' convergence for sequences of functions.

Random Attractor Family for the Kirchhoff Equation of Higher Order with White Noise

The existence of random attractor family for a class of nonlinear high-order Kirchhoff equation stochastic dynamical systems with white noise is studied. The Ornstein-Uhlenbeck process and the weak solution of the equation are used to deal with the stochastic terms. The equation is transformed into a general stochastic equation. The bounded stochastic absorption set is obtained by estimating the solution of the equation and the existence of the random attractor family is obtained by isomorphic mapping method. Temper random compact sets of random attractor family are obtained.

Asymptotic behavior of some hitting probabilities for sums of IID Gaussian random sets

ABSTRACTIn this article, we consider sequences of partial sums of iid Gaussian random sets (with respect to the Minkowski sum) and we study the asymptotic behavior of some hitting probabilities (of suitable sets of ) for these partial sums. We also illustrate the use of the importance sampling for the estimation of these hitting probabilities by Monte Carlo simulations. We obtain the analog of well-known results for level crossing probabilities of random walks, and we refer to a version of the classical Cramér's Theorem in large deviations for random compact sets existing in the literature.

Existence of random invariant periodic curves via random semiuniform ergodic theorem

We employ an extension of ergodic theory to the random setting to investigate the existence of random periodic solutions of random dynamical systems. Given that a random dynamical system on a cylinder [Formula: see text] has a dissipative structure, we proved that a random invariant compact set can be expressed as a union of finite of number of random periodic curves. The idea in this paper is closely related to the work recently considered by Zhao and Zheng [46].

Asymptotic tests for interval-valued means

We develop asymptotic tests for the means of interval-valued population. The problem is formulated and studied in the framework of random compact convex sets. Under both one-sample and two-sample settings, we derive analytical forms of the probability density functions for the limiting null distributions. Large sample testing rules are given based on numerical integration of the Bessel function.

Towards a mean body for apparel design

This paper focuses on shape average with applications to the apparel industry. Apparel industry uses a consensus sizing system; its major concern is to fit most of the population into it. Since anthropometric measures do not grow linearly, it is important to find prototypes to accurately represent each size. This is done using random compact mean sets, obtained from a cloud of 3D points given by a scanner and applying to the sample a previous definition of mean set. Additionally, two approaches to define confidence sets are introduced. The methodology is applied to data obtained from a real anthropometric survey.

Multivariate Poisson distributions associated with Boolean models

We consider a d-dimensional Boolean model $$\varXi = (\varXi _1+X_1)\cup (\varXi _2+X_2)\cup \cdots $$ generated by a Poisson point process $$\{X_i, i\ge 1\}$$ with intensity measure $$\varLambda $$ and a sequence $$\{\varXi _i, i\ge 1\}$$ of independent copies of some random compact set $$\varXi _0\,$$ . Given compact sets $$K_1,\ldots ,K_{\ell }$$ , we show that the discrete random vector $$(N(K_1),\ldots ,N(K_\ell ))$$ , where $$N(K_j)$$ equals the number of shifted sets $$\varXi _i+X_i$$ hitting $$K_j$$ , obeys an $$\ell $$ -variate Poisson distribution with $$2^{\ell }-1$$ parameters. We obtain explicit formulae for all these parameters which can be estimated consistently from an observation of the union set $$\varXi $$ in some unboundedly expanding window $$W_n$$ (as $$n \rightarrow \infty $$ ) provided that the Boolean model is stationary. Some of these results can be extended to unions of Poisson k-cylinders for $$1\le k < d$$ and more general set-valued functionals of independently marked Poisson processes.

Fractal dimension of random invariant sets for nonautonomous random dynamical systems and random attractor for stochastic damped wave equation

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.