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About 1,379 papers for “Random compact set”

Reachability analysis for linear hybrid set-dynamics driven by random convex compact sets

Ion Matei, John S. Baras · 2017

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.

Cited by 0

A Theorem on Continuity of a Random Function on a Compact Set in Hilbert Space

A. V. Skorokhod · Theory of Probability and Its Applications · 1974

Previous article Next article A Theorem on Continuity of a Random Function on a Compact Set in Hilbert SpaceA. V. SkorokhodA. V. Skorokhodhttps://doi.org/10.1137/1118097PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] I. I. Gikhman and , A. V. Skorohod, The theory of stochastic processes. I, Springer-Verlag, New York, 1974viii+570 MR0346882 (49:11603) 0291.60019 CrossrefGoogle Scholar[2] M. I. Jadrenko, The continuity of the sample functions of Gaussian random fields on Hilbert space, Dopovīdī Akad. Nauk Ukraïn. RSR Ser. A, 1968 (1968), 734–736, (In Russian.) MR0243598 (39:4919) 0167.16901 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Volume 18, Issue 4| 1974Theory of Probability & Its Applications History Submitted:05 March 1973Published online:28 July 2006 InformationCopyright © 1974 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1118097Article page range:pp. 771-774ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Cited by 1

On logarithmically optimal exact simulation of max-stable and related random fields on a compact set

Zhipeng Liu, José Blanchet, A. B. Dieker et al. · Bernoulli · 2019

We consider the random field \begin{equation*}M(t)=\mathop{\mathrm{sup}}_{n\geq1}\{-\log A_{n}+X_{n}(t)\},\qquad t\in T,\end{equation*} for a set $T\subset\mathbb{R}^{m}$, where $(X_{n})$ is an i.i.d. sequence of centered Gaussian random fields on $T$ and $0<A_{1}<A_{2}<\cdots$ are the arrivals of a general renewal process on $(0,\infty)$, independent of $(X_{n})$. In particular, a large class of max-stable random fields with Gumbel marginals have such a representation. Assume that one needs $c(d)=c(\{t_{1},\ldots,t_{d}\})$ function evaluations to sample $X_{n}$ at $d$ locations $t_{1},\ldots,t_{d}\in T$. We provide an algorithm which samples $M(t_{1}),\ldots,M(t_{d})$ with complexity $O(c(d)^{1+o(1)})$ as measured in the $L_{p}$ norm sense for any $p\ge1$. Moreover, if $X_{n}$ has an a.s. converging series representation, then $M$ can be a.s. approximated with error $\delta$ uniformly over $T$ and with complexity $O(1/(\delta\log(1/\delta))^{1/\alpha})$, where $\alpha$ relates to the Hölder continuity exponent of the process $X_{n}$ (so, if $X_{n}$ is Brownian motion, $\alpha=1/2$).

Cited by 9Open access

Some Properties of the Prolongation Limit Random Sets in Random Dynamical Systems

Sundus Talib Mohsin, Ihsan Jabbar Kadhim · Journal of Al-Qadisiyah for Computer Science and Mathematics · 2019

The aim of this paper is to study the omega limit set with new concepts of the prolongation limit random sets in random dynamical systems, where some properties are proved and introduced such as the relation among the orbit closure, orbit and omega limit random set. Also we prove that the first prolongation of a closed random set containing this set, the first prolongation is closed and invariant. In addition, it is connected whenever it is compact provided that the phase space of the random dynamical systems is locally compact. Then, we study the prolongational limit random set and examined some essential properties of this set. Finally, the relation among the first prolongation, the prolongational limit random set and the positive trajectory of a random set is given and proved.

Cited by 1Open access

Solidification of porous interfaces and disconnection

Maximilian Nitzschner, Alain-Sol Sznitman · Journal of the European Mathematical Society · 2020

In this article we obtain uniform estimates on the absorption of Brownian motion by porous interfaces surrounding a compact set. An important ingredient is the construction of certain resonance sets, which are hard to avoid for Brownian motion starting in the compact set. As an application of our results, we substantially strengthen the results of [22], and obtain when d \geq 3 , large deviation upper bounds on the probability that simple random walk in \mathbb Z^d , or random interlacements in \mathbb Z^d , when their vacant set is in a strongly percolative regime, disconnect the discrete blow-up of a regular compact set from the boundary of the discrete blow-up of a box containing the compact set in its interior. Importantly, we make no convexity assumption on the compact set. It is plausible, although open at the moment, that the upper bounds that we derive in this work match in principal order the lower bounds of [15] in the case of random interlacements, and of [14] for the simple random walk.

Cited by 15Open access

ATTRACTORS FOR STOCHASTIC LATTICE DYNAMICAL SYSTEMS

Peter W. Bates, Hannelore Lisei, Kening Lu · Stochastics and Dynamics · 2006

We consider a one-dimensional lattice with diffusive nearest neighbor interaction, a dissipative nonlinear reaction term and additive independent white noise at each node. We prove the existence of a compact global random attractor within the set of tempered random bounded sets. An interesting feature of this is that, even though the spatial domain is unbounded and the solution operator is not smoothing or compact, pulled back bounded sets of initial data converge under the forward flow to a random compact invariant set.

Cited by 311

Solidification of porous interfaces and disconnection

Nitzschner, Maximilian, Sznitman, Alain-Sol · arXiv (Cornell University) · 2017

In this article we obtain uniform estimates on the absorption of Brownian motion by porous interfaces surrounding a compact set. An important ingredient is the construction of certain resonance sets, which are hard to avoid for Brownian motion starting in the compact set. As an application of our results, we substantially strengthen the results of arXiv:1412.3960, and obtain when $d \ge 3$, large deviation upper bounds on the probability that simple random walk in $Z^d$, or random interlacements in $Z^d$, when their vacant set is in a strongly percolative regime, disconnect the discrete blow-up of a regular compact set from the boundary of the discrete blow-up of a box containing the compact set in its interior. Importantly, we make no convexity assumption on the compact set. It is plausible, although open at the moment, that the upper bounds that we derive in this work match in principal order the lower bounds of Xinyi Li and the second author (see arXiv:1310.2177) in the case of random interlacements, and of Xinyi Li (see arXiv:1412.3959) for the simple random walk.

Cited by 0Open access

(2) If dimH F&amp;gt; d

Davar Khoshnevisan, Yimin Xiao · 2011

Abstract. An N-parameter Brownian sheet in Rd maps a non-random compact set F in RN + to the random compact set B(F) in Rd. We prove two results on the image-set B(F): (1) It has positive d-dimensional Lebesgue measure if and only if F has positive d-dimensional capacity. This generalizes greatly the earlier works of

Cited by 0

Characterization and Domains of Attraction of $p$-Stable Random Compact Sets

Evarist Giné, Marjorie G. Hahn · The Annals of Probability · 1985

Let $(\mathscr{K} (\mathbb{B}), \delta)$ denote the nonempty compact subsets of a separable Banach space $\mathbb{B}$ topologized by the Hausdorff metric. Let $K, K_1, K_2$ be i.i.d. random compact convex sets in $\mathbb{B}. K$ is called $p$-stable if for each $\alpha, \beta \geq 0$ there exist compact convex sets $C$ and $D$ such that $\mathscr{L}(\alpha K_1 + \beta K_2 + C) = \mathscr{L}((\alpha^p + \beta^p)^{1/p}K + D)$ where + refers to Minkowski sum. A characterization of the support function for a compact convex set is provided and then utilized to determine all $p$-stable random compact convex sets. If $1 \leq p \leq 2$, they are trivial, merely translates of a fixed compact convex set by a $p$-stable $\mathbb{B}$-valued random variable. For $0 < p < 1$, they are translates of stochastic integrals with respect to nonnegative independently scattered $p$-stable measures on the unit ball of $\operatorname{co} \mathscr{K}(\mathbb{B})$. Deconvexification is also discussed. The domains of attraction of $p$-stable random compact convex sets with $0 < p < 1$ are completely characterized. The case $1 < p \leq 2$ is considered in Gine, Hahn and Zinn (1983). Precedents: Lyashenko (1983) and Vitale (1983) characterize the Gaussian random compact sets in $\mathbb{R}^d$.

Cited by 29Open access

Testing equality in distribution of random convex compact sets via theory of N-distances and its application to assessing similarity of general random sets

Vesna Gotovac, Kateřina Helisová · arXiv (Cornell University) · 2018

This paper concerns a method of testing equality of distribution of random convex compact sets and the way how to use the test to distinguish between two realisations of general random sets. The family of metrics on the space of distributions of random convex compact sets is constructed using the theory of N-distances and characteristic functions of random convex compact sets. Further, the approximation of the metrics through its finite dimensional counterparts is proposed, which lead to a new statistical test for testing equality in distribution of two random convex compact sets. Then, it is described how to approximate a realisation of a general random set by a union of convex compact sets, and it is shown how to determine whether two realisations of general random sets come from the same process using the constructed test. The procedure is justified by an extensive simulation study.

Cited by 1Open access

Real and complex zeros of Riemannian random waves

Steve Zelditch · Contemporary mathematics - American Mathematical Society · 2009

We consider Riemannian random waves, i.e. Gaussian random linear combination of eigenfunctions of the Laplacian on a compact Riemannian manifold with frequencies from a short interval (`asymptotically fixed frequency'). We first show that the expected limit distribution of the real zero set of a is uniform with respect to the volume form of a compact Riemannian manifold $(M, g)$. We then show that the complex zero set of the analytic continuations of such Riemannian random waves to a Grauert tube in the complexification of $M$ tends to a limit current.

Cited by 55

Pullback attractors for multi-valued non-compact random dynamical systems generated by semi-linear degenerate parabolic equations with unbounded delays

Jingyu Wang, Yejuan Wang, Dun Zhao · Stochastics and Dynamics · 2015

The theory of pullback attractors for multi-valued non-compact random dynamical systems and a method of asymptotic compactness based on the concepts of the Kuratowski measure of the non-compactness of a bounded set are used to prove the existence of pullback attractors for the multi-valued non-compact random dynamical systems associated with the semi-linear degenerate parabolic unbounded delay equations with both deterministic and random external terms.

Cited by 11

Stability and Asymptotic Stability of Closed Random Sets

Ihsan Jabbar Kadhim, Mohammed Shanan Imran · Journal of Physics Conference Series · 2020

Abstract In this paper the concepts of Stability, Asymptotic Stability of Closed Random Sets family in (deterministic) dynamical system are defined in (random) dynamical systems. Also some main result such as uniformly stable, compact asymptotically stable set are proved in random dynamical system.

Cited by 1Open access

A Random Attractor Family of the High Order Beam Equations with White Noise

Guoguang Lin, Jie Liu · International Journal of Modern Nonlinear Theory and Application · 2020

In this paper, we studied a class of damped high order Beam equation stochastic dynamical systems with white noise. First, the Ornstein-Uhlenbeck process is used to transform the equation into a noiseless random equation with random variables as parameters. Secondly, by estimating the solution of the equation, we can obtain the bounded random absorption set. Finally, the isomorphism mapping method and compact embedding theorem are used to obtain the system. It is progressively compact, then we can prove the existence of random attractors.

Cited by 1Open access

The Krein-Milman 定理 in random locally convex modules and its applications

Tiexin Guo, Wang Yachao, Yan Tang · Scientia Sinica Mathematica · 2023

This paper first deeply studies stably compact sets of a random locally convex module by proving that they are complete with respect to the $(\varepsilon,\lambda)$-topology $\mathcal{T}_{\varepsilon,\lambda}$ and characterizing them in the way that each $\sigma$-stable family of $\sigma$-stable $\mathcal{T}_{\varepsilon,\lambda}$ -closed subsets of them with the finite intersection 性质 has a nonempty intersection. Based on the preliminaries, we further give the Weierstrass 定理 for a proper lower semicontinuous $\bar{L}^0$-valued $\sigma$-stable function defined on a stably compact set, which implies that a stably compact $L^0$-convex set must be $L^0$-convexly compact. Then, we introduce the notion of an $L^0$-extreme point for an $L^0$-convex set and prove the corresponding Krein-Milman 定理 for an $L^0$-convexly compact set. Besides, some interesting connection and comparison between the generalized Krein-Milman 定理 and the classical one for a compact convex set in a locally convex space are also given. Finally, as an application of the generalized Krein-Milman 定理, we prove that a proper, lower semicontinuous, $\sigma$-stable and $L^0$-quasiconvex function on an $L^0$-convexly compact set of a random locally convex module can attain its minimum value, and further if in addition, $f$ is $L^0(\mathcal{F})$-valued and $L^0$-affine, then $f$ can attain its minimum value at some $L^0$-extreme point of the $L^0$-convexly compact set.

Cited by 1Open access

Fractals associated with random multiplication of polynomials

Skordev, G. (Bremen Univ. (Germany)), Mauldin, R.D. (North Texas State Univ., Denton, TX (United States). Dept. of Mathematics), Bremen Univ. (Germany). Inst. fuer Dynamische Systeme · OpenGrey (Institut de l'Information Scientifique et Technique) · 1998

Random multiplication of a given set of s polynomials with coefficients in a finite field following a random sequence generated by Bernoulli trial with s possible outcomes is a (time-dependent) linear cellular automaton (LCA). As in the case of LCA with states in a finite field we associate with this sequence a compact set - the rescaled evolution set. The law of the iterated logarithm implies that this fractal set almost surely does not depend on the random sequence. (orig.)

Cited by 0

Bounds for Restricted Selections of Random Sets

Arie Beresteanu, Behrooz Moosavi Rameznzadeh · ArXiv.org · 2025

We study constrained selection sets of random closed sets defined on a non-atomic probability space. Given a random interval $Y=[y_L,y_U]$ and scalar constraints on the expectation or the median of admissible selections, we characterize the restricted selection set and establish sharp bounds on the attainable ranges of means, medians, and event probabilities. In particular, we give conditions under which every value in the Aumann expectation range is realized as the mean of a measurable selection, and we obtain explicit formulas for the extremal expectations under median and higher-moment restrictions via rearrangement and convex-duality arguments. We further show that the selection set of any random compact convex set in $\R^d$ can be approximated in $L^1$ by selection sets of disjoint unions of random cubes, each of which decomposes coordinate-wise into one-dimensional interval selection problems. This gives us an approximation-based reduction of constrained selection problems for random compact convex sets in $\R^d$.

Cited by 0Open access

UNIFORM LYAPUNOV EXPONENTS ARE REALISED BY ERGODIC INVARIANT MEASURES

Hans Crauel · Stochastics and Dynamics · 2001

Given a differentiable random dynamical system on a finite-dimensional differentiable manifold with a compact random invariant set, we show that there exists an ergodic invariant measure, supported by the invariant set, such that the leading Lyapunov exponent associated with this invariant measure equals the uniform Lyapunov exponent with respect to the invariant set. This is extended to sums of Lyapunov exponents and to the Lyapunov dimension of the set.

Cited by 1