Random differential inclusions with convex right hand sides

Krystyna Grytczuk; Emilia Rotkiewicz

Displaying similar documents to “Random differential inclusions with convex right hand sides”

Aging and quenched localization for one-dimensional random walks in random environment in the sub-ballistic regime

Nathanaël Enriquez, Christophe Sabot, Olivier Zindy (2009)

Bulletin de la Société Mathématique de France

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We consider transient one-dimensional random walks in a random environment with zero asymptotic speed. An aging phenomenon involving the generalized Arcsine law is proved using the localization of the walk at the foot of “valleys“ of height $log t$ . In the quenched setting, we also sharply estimate the distribution of the walk at time $t$ .

Slowdown estimates and central limit theorem for random walks in random environment

Alain-Sol Sznitman (2000)

Journal of the European Mathematical Society

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This work is concerned with asymptotic properties of multi-dimensional random walks in random environment. Under Kalikow’s condition, we show a central limit theorem for random walks in random environment on $ℤ^{d}$ , when $d > 2$ . We also derive tail estimates on the probability of slowdowns. These latter estimates are of special interest due to the natural interplay between slowdowns and the presence of traps in the medium. The tail behavior of the renewal time constructed in [25] plays an important...

On the existence and asymptotic behavior of the random solutions of the random integral equation with advancing argument

Henryk Gacki (1996)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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1. Introduction Random Integral Equations play a significant role in characterizing of many biological and engineering problems [4,5,6,7]. We present here new existence theorems for a class of integral equations with advancing argument. Our method is based on the notion of a measure of noncompactness in Banach spaces and the fixed point theorem of Darbo type. We shall deal with random integral equation with advancing argument $x (t, ω) = h (t, ω) + \int^{t + δ (t)} ₀ k (t, τ, ω) f (τ, x_{τ} (ω)) d τ$ , (t,ω) ∈ R⁺ × Ω, (1) where (i) (Ω,A,P) is a complete probability...

Random fixed points for a certain class of asymptotically regular mappings

Balwant Singh Thakur, Jong Soo Jung, Daya Ram Sahu, Yeol Je Cho (1998)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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Let (Ω, σ) be a measurable space and K a nonempty bounded closed convex separable subset of a p-uniformly convex Banach space E for p > 1. We prove a random fixed point theorem for a class of mappings T:Ω×K ∪ K satisfying the condition: For each x, y ∈ K, ω ∈ Ω and integer n ≥ 1, ⃦Tⁿ(ω,x) - Tⁿ(ω,y) ⃦ ≤ aₙ(ω)· ⃦x - y ⃦ + bₙ(ω) ⃦x -Tⁿ(ω,x) ⃦ + ⃦y - Tⁿ(ω,y) ⃦ + cₙ(ω) ⃦x - Tⁿ(ω,y) ⃦ + ⃦y - Tⁿ(ω,x) ⃦, where aₙ, bₙ, cₙ: Ω → [0, ∞) are functions satisfying certain conditions and Tⁿ(ω,x)...

Excited against the tide: a random walk with competing drifts

Mark Holmes (2012)

Annales de l'I.H.P. Probabilités et statistiques

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We study excited random walks in i.i.d. random cookie environments in high dimensions, where the $k$ th cookie at a site determines the transition probabilities (to the left and right) for the $k$ th departure from that site. We show that in high dimensions, when the expected right drift of the first cookie is sufficiently large, the velocity is strictly positive, regardless of the strengths and signs of subsequent cookies. Under additional conditions on the cookie environment, we show that...

Strong laws of large numbers for sequences of blockwise and pairwise $m$ -dependent random variables in metris spaces

Nguyen Van Quang, Pham Tri Nguyen (2016)

Applications of Mathematics

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The aim of the paper is to establish strong laws of large numbers for sequences of blockwise and pairwise $m$ -dependent random variables in a convex combination space with or without compactly uniformly integrable condition. Some of our results are even new in the case of real random variables.

Random fixed points of increasing compact random maps

Ismat Beg (2001)

Archivum Mathematicum

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Let $(Ω, Σ)$ be a measurable space, $(E, P)$ be an ordered separable Banach space and let $[a, b]$ be a nonempty order interval in $E$ . It is shown that if $f : Ω \times [a, b] \to E$ is an increasing compact random map such that $a \leq f (ω, a)$ and $f (ω, b) \leq b$ for each $ω \in Ω$ then $f$ possesses a minimal random fixed point $α$ and a maximal random fixed point $β$ .

Set-valued random differential equations in Banach space

Mariusz Michta (1995)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We consider the problem of the existence of solutions of the random set-valued equation: (I) $D_{H} X_{t} = F (t, X_{t}) P . 1$ , t ∈ [0,T] -a.e.; X₀ = U p.1 where F and U are given random set-valued mappings with values in the space $K_{c} (E)$ , of all nonempty, compact and convex subsets of the separable Banach space E. Under certain restrictions on F we obtain existence of solutions of the problem (I). The connections between solutions of (I) and solutions of random differential inclusions are investigated.

On the limiting velocity of random walks in mixing random environment

Xiaoqin Guo (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider random walks in strong-mixing random Gibbsian environments in $ℤ^{d}$ , $d \geq 2$ . Based on regeneration arguments, we will first provide an alternative proof of Rassoul-Agha’s conditional law of large numbers (CLLN) for mixing environment ( (2005) 36–44). Then, using coupling techniques, we show that there is at most one nonzero limiting velocity in high dimensions ( $d \geq 5$ ).

Ballistic regime for random walks in random environment with unbounded jumps and Knudsen billiards

Francis Comets, Serguei Popov (2012)

Annales de l'I.H.P. Probabilités et statistiques

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We consider a random walk in a stationary ergodic environment in $ℤ$ , with unbounded jumps. In addition to uniform ellipticity and a bound on the tails of the possible jumps, we assume a condition of strong transience to the right which implies that there are no “traps.” We prove the law of large numbers with positive speed, as well as the ergodicity of the environment seen from the particle. Then, we consider Knudsen stochastic billiard with a drift in a random tube in $ℝ^{d}$ , $d \geq 3$ , which serves...

Slowdown estimates for ballistic random walk in random environment

Noam Berger (2012)

Journal of the European Mathematical Society

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We consider models of random walk in uniformly elliptic i.i.d. random environment in dimension greater than or equal to 4, satisfying a condition slightly weaker than the ballisticity condition $(T^{'})$ . We show that for every $ϵ > 0$ and $n$ large enough, the annealed probability of linear slowdown is bounded from above by $exp (- {(log n)}^{d - ϵ})$ . This bound almost matches the known lower bound of $exp (- C {(log n)}^{d})$ , and signiﬁcantly improves previously known upper bounds. As a corollary we provide almost sharp estimates for the quenched...

Coherent randomness tests and computing the $K$ -trivial sets

Laurent Bienvenu, Noam Greenberg, Antonín Kučera, André Nies, Dan Turetsky (2016)

Journal of the European Mathematical Society

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We introduce Oberwolfach randomness, a notion within Demuth’s framework of statistical tests with moving components; here the components’ movement has to be coherent across levels. We show that a ML-random set computes all $K$ -trivial sets if and only if it is not Oberwolfach random, and indeed that there is a $K$ -trivial set which is not computable from any Oberwolfach random set. We show that Oberwolfach random sets satisfy effective versions of almost-everywhere theorems of analysis,...

The Infinite Random Order of Dimension $k$

Peter Winkler (1985)

Publications du Département de mathématiques (Lyon)

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Weak quenched limiting distributions for transient one-dimensional random walk in a random environment

Jonathon Peterson, Gennady Samorodnitsky (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We consider a one-dimensional, transient random walk in a random i.i.d. environment. The asymptotic behaviour of such random walk depends to a large extent on a crucial parameter $κ g t; 0$ that determines the fluctuations of the process. When $0 l t; κ l t; 2$ , the averaged distributions of the hitting times of the random walk converge to a $κ$ -stable distribution. However, it was shown recently that in this case there does not exist a quenched limiting distribution of the hitting times. That is, it is not true...

Random fixed point theorems for a certain class of mappings in Banach spaces

Jong Soo Jung, Yeol Je Cho, Shin Min Kang, Byung-Soo Lee, Balwant Singh Thakur (2000)

Czechoslovak Mathematical Journal

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Let $(Ω, Σ)$ be a measurable space and $C$ a nonempty bounded closed convex separable subset of $p$ -uniformly convex Banach space $E$ for some $p > 1$ . We prove random fixed point theorems for a class of mappings $T Ω \times C \to C$ satisfying: for each $x, y \in C$ , $ω \in Ω$ and integer $n \geq 1$ , $∥ T^{n} (ω, x) - T^{n} (ω, y) ∥ \leq a (ω) \cdot ∥ x - y ∥ + b (ω) {∥ x - T^{n} (ω, x) ∥ + ∥ y - T^{n} (ω, y) ∥} + c (ω) {∥ x - T^{n} (ω, y) ∥ + ∥ y - T^{n} (ω, x) ∥},$ where $a, b, c Ω \to [0, \infty)$ are functions satisfying certain conditions and $T^{n} (ω, x)$ is the value at $x$ of the $n$ -th iterate of the mapping $T (ω, \cdot)$ . Further we establish for these mappings some random fixed point theorems in a Hilbert space, in $L^{p}$ spaces, in Hardy spaces $H^{p}$ and in Sobolev...

Tail and moment estimates for sums of independent random variables with logarithmically concave tails

E. Gluskin, S. Kwapień (1995)

Studia Mathematica

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For random variables $S = \sum_{i = 1}^{\infty} α_{i} ξ_{i}$ , where $(ξ_{i})$ is a sequence of symmetric, independent, identically distributed random variables such that $l n P (| ξ_{i} | \geq t)$ is a concave function we give estimates from above and from below for the tail and moments of S. The estimates are exact up to a constant depending only on the distribution of ξ. They extend results of S. J. Montgomery-Smith [MS], M. Ledoux and M. Talagrand [LT, Chapter 4.1] and P. Hitczenko [H] for the Rademacher sequence.