Oscillation criteria for nonlinear second order differential equations with damping

Summary Using the integral average method, we give some new oscillation criteria for the second order differential equation with damped term (a(t)Ψ(x(t))K(x'(t)))'+p(t)K(x'(t))+q(t)f(x(t))=0, t<span style='font-size:10.0pt; font-family:"Lucida Sans Unicode"'>≧t₀. These results improve and generalize the oscillation criteria in<span lang=EN-US style='font-size:10.0pt;mso-ansi-language:EN-US'>[1], because they eliminate both the differentiability of p(t) and the sign of p(t), q(t). As a consequence, improvements of Sobol's type oscillation criteria are obtained.     

Asymptotic convergence of nonlinear contraction semigroups in Hilbert space

Let S be a contraction semigroup on a closed convex subset C of a Hilbert space. If the generator of S satisfies a strengthened monotonicity condition then the weak lim_{t → ∞}S(t)x exists for all x in C. As one consequence, the method of steepest descent converges weakly for convex functions in Hilbert space; and it converges strongly for even convex functions.     

Clustering of linearly interacting diffusions and universality of their long-time limit distribution

Let K⊂ℝ ^d (d≥ 1) be a compact convex set and Λ a countable Abelian group. We study a stochastic process X in K ^Λ, equipped with the product topology, where each coordinate solves a SDE of the form dX _i (t) = ∑ _j a(j−i) (X _j (t) −X _i (t))dt + σ (X _i (t))dB _i (t). Here a(·) is the kernel of a continuous-time random walk on Λ and σ is a continuous root of a diffusion matrix w on K. If X(t) converges in distribution to a limit X(∞) and the symmetrized random walk with kernel a _S (i) = a(i) + a(−i) is recurrent, then each component X _i (∞) is concentrated on {x∈K : σ(x) = 0 and the coordinates agree, i.e., the system clusters. Both these statements fail if a _S is transient. Under the assumption that the class of harmonic functions of the diffusion matrix w is preserved under linear transformations of K, we show that the system clusters for all spatially ergodic initial conditions and we determine the limit distribution of the components. This distribution turns out to be universal in all recurrent kernels a _S on Abelian groups Λ. Received: 10 May 1999 / Revised version: 18 April 2000 / Published online: 22 November 2000     

Strongly and weakly almost periodic linear maps on semigroup algebras

Let S be a foundation locally compact topological semigroup. Two new topologies τ _c and τ _w are introduced on M _a (S)^*. We introduce τ _c and τ _w almost periodic functionals in M _a (S)^*. We study these classes and compare them with each other and with the norm almost periodic and weakly almost periodic functionals. For f∈M _a (S)^*, it is proved that T _f ∈ℬ(M _a (S),M _a (S)^*) is strong almost periodic if and only if f is τ _c -almost periodic. Indeed, we have obtained a generalization of a well known result of Crombez for locally compact group to a more general setting of foundation topological semigroups. Finally if P(S) (the set of all probability measures in M _a (S)) has the semiright invariant isometry property, it is shown that the set of τ _w -almost periodic functionals has a topological left invariant mean.     

Spectral mapping theorems for exponentially bounded c-semigroups in Banach spaces

LetZ be a generator of an exponentially boundedC-semigroup {S _t } _t≥0 in a Banach space and letT _t =C ⁻¹ S _t . We show that the spectral mapping theorems such as exp(tσ(Z)) ⊂ σ(T _t ) and exp(tσ _p (Z)) ⊂ tσ _p (T _t ) ⊂ exp(tσ _p (Z)) ⋃ {0} for everyt≥0 hold. The present studies were supported by the Basic Science Research Institute Program, Ministry of Education, 1987.     

Random ergodic theorems and real cocycles

We study mean convergence of ergodic averages associated to a measure-preserving transformation or flow τ along the random sequence of times κ _n (ω) given by the Birkhoff sums of a measurable functionF for an ergodic measure-preserving transformationT. We prove that the sequence (k _n(ω)) is almost surely universally good for the mean ergodic theorem, i.e., that, for almost every, ω, the averages (*) converge for every choice of τ, if and only if the “cocycle”F satisfies a cohomological condition, equivalent to saying that the eigenvalue group of the “associated flow” ofF is countable. We show that this condition holds in many natural situations. When no assumption is made onF, the random sequence (k _n(ω)) is almost surely universally good for the mean ergodic theorem on the class of mildly mixing transformations τ. However, for any aperiodic transformationT, we are able to construct an integrable functionF for which the sequence (k _n(ω)) is not almost surely universally good for the class of weakly mixing transformations.     

A class of risk processes with reserve-dependent premium rate: sample path large deviations and importance sampling

Let (X(t)) be a risk process with reserve-dependent premium rate, delayed claims and initial capital u. Consider a class of risk processes {(X ^ε (t)): ε > 0} derived from (X(t)) via scaling in a slow Markov walk sense, and let Ψ_ε(u) be the corresponding ruin probability. In this paper we prove sample path large deviations for (X ε (t)) as ε → 0. As a consequence, we give exact asymptotics for log Ψ_ε(u) and we determine a most likely path leading to ruin. Finally, using importance sampling, we find an asymptotically efficient law for the simulation of Ψ_ε(u). AMS Subject Classifications 60F10, 91B30 This work has been partially supported by Murst Project “Metodi Stocastici in Finanza Matematica”     

A quantum spin system with random interactions I

We study a quantum spin glass as a quantum spin system with random interactions and establish the existence of a family of evolution groups {τ_t(ω)}_ω∈/Ω of the spin system. The notion of ergodicity of a measure preserving group of automorphisms of the probability space Ω, is used to prove the almost sure independence of the Arveson spectrum Sp(τ(ω)) of τ_t(ε). As a consequence, for any family of (τ(ω),β) — KMS states {ρ(ω)}, the spectrum of the generator of the group of unitaries which implement τ(ω) in the GNS representation is also almost surely independent of ω.     

Proprietes generiques de processus croises

LetS _φ be the skew product transformation(x, g)↦(Sx, gφ(x)) defined on Ω×G, where Ω is a compact metric space,G a compact metric group with its Haar measureh. IfS is a μ-continuous transformation where μ is a Borel measure on Ω, ergodic with respect toS, we study the setE ₀ of μ-continuous applications φ:Ω→G such that μ⩀h is ergodic (with respect toS _φ). For example,E ₀ is residual in the group of μ-continuous applications from Ω toG with the uniform convergence topology. We also study the weakly mixing case. Some arithmetic applications are given.     

Some limit results for probabilities estimates of Brownian motion with polynomial drift

Let (B _t + f(t)) _t∈[0,+∞) be a Brownian motion with polynomial drift f(t), where f(t) is a polynomial. Some Limit Results for Lower tail and large deviation probabilities estimates, and Level crossing probabilities estimates of (B _t + f(t)) _t∈[0,+∞) are given in this paper.     

On the Asymptotic Behavior of Perturbed Strongly Continuous Semigroups

For a strongly continuous semigroup (T(t))_t≥0 with generator A on a Banach space X and an A–bounded perturbation B we characterize norm continuity and compactness of the terms in the Dyson–Phillips series of the perturbed semigroup (S(t))_t≥0 .This allows us to characterize uniform exponential stability of (S(t))_t≥0 by spectral conditions on (T(t))_t≥0 and A + B. The results are applied to a delay differential equation.     

Asymptotic results on infinite tandem queueing networks

We consider an infinite tandem queueing network consisting of ·/GI/1/∞ stations with i.i.d. service times. We investigate the asymptotic behavior of t(n, k), the inter-arrival times between customers n and n + 1 at station k, and that of w(n, k), the waiting time of customer n at station k. We establish a duality property by which w(n, k) and the “idle times”y(n, k) play symmetrical roles. This duality structure, interesting by itself, is also instrumental in proving some of the ergodic results. We consider two versions of the model: the quadrant and the half-plane. In the quadrant version, the sequences of boundary conditions {w(0,k), k∈ℕ} and {t(n, 0), n∈ℕ}, are given. In the half-plane version, the sequence {t(n, 0), n∈ℕ} is given. Under appropriate assumptions on the boundary conditions and on the services, we obtain ergodic results for both versions of the model. For the quadrant version, we prove the existence of temporally ergodic evolutions and of spatially ergodic ones. Furthermore, the process {t(n, k), n∈ℕ} converges weakly with k to a limiting distribution, which is invariant for the queueing operator. In the more difficult half plane problem, the aim is to obtain evolutions which are both temporally and spatially ergodic. We prove that 1/n∑ _k=1 ⁿ w(0, k) converges almost surely and in L ₁ to a finite constant. This constitutes a first step in trying to prove that {t(n,k), n∈ℤ} converges weakly with k to an invariant limiting distribution. Received: 23 March 1999 / Revised version: 5 January 2000 / Published online: 12 October 2000     

Invariant Measure for the Markov Process Corresponding to a PDE System

In this paper, we consider the Markov process （X^∈（t）, Z^∈（t）） corresponding to a weakly coupled elliptic PDE system with a small parameter ∈ 〉 0. We first prove that （X^∈（t）, Z^∈（t）） has the Feller continuity by the coupling method, and then prove that （X^∈（t）, Z^∈（t）） has an invariant measure μ^∈（·） by the Foster-Lyapunov inequality. Finally, we establish a large deviations principle for μ^∈（·） as the small parameter e tends to zero.     

Propriétés Régularisantes de Certains Semi-Groupes Non Linéaires

Let φ be a convex l.s.c. function fromH (Hilbert) into ] - ∞, ∞ ] andD(φ)={u ∈H; φ(u)<+∞}. It is proved that for everyu ₀ ∈D(φ) the equation − (du/dt)(t ∈ ∂φ(u(t)),u(0)=u ₀ has a solution satisfying ÷(du(t)/dt)÷ ≦(c ₁/t)+c ₂. The behavior ofu(t) in the neighborhood oft=0 andt=+∞ as well as the inhomogeneous equation (du(t)/dt)+∂φ(u(t)) ∈f(t) are then studied. Solutions of some nonlinear boundary value problems are given as applications.      

Multiparameter inverse problem of approximation by functions with given supports

Let L _p (S), 0 < p < +∞, be a Lebesgue space of measurable functions on S with ordinary quasinorm ∥·∥ _p . For a system of sets {B _t |t ∈ [0, +∞) ⁿ } and a given function ψ: [0, +∞) ⁿ ↦ [ 0, +∞), we establish necessary and sufficient conditions for the existence of a function f ∈ L _p (S) such that inf {∥f − g∥ _p ^p g ∈ L _p (S), g = 0 almost everywhere on S\B _t } = ψ (t), t ∈ [0, +∞) ⁿ . As a consequence, we obtain a generalization and improvement of the Dzhrbashyan theorem on the inverse problem of approximation by functions of the exponential type in L ₂. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 8, pp. 1116–1127, August, 2006.     

A recurrence property of smooth functions

LetP andQ be real polynomials of degreesd ande, respectively, andf a periodic function. It is shown that, iff iss times differentiable atQ(0), wheres≧7de ³ log 14e ³, then for every ɛ>0 the diophantine inequality ≧FF5C;P(x)f(Q(x)) -P(0)f(Q(0)) -y≧ εx≠0, has a solution. This settles in particular a question raised by Furstenberg and Weiss [6].     

On the spectral mapping theorem for perturbed strongly continuous semigroups

We consider a strongly continuous semigroup (T(t))_{t \geqq 0}(T(t))_{t \geqq 0} with generator A on a Banach space X, an A-bounded perturbation B, and the semigroup (S(t))_{t \geqq 0}(S(t))_{t \geqq 0} generated by A + B. Using the critical spectrum introduced recently, we improve existing spectral mapping theorems for the perturbed semigroup (S(t))_{t \geqq 0}(S(t))_{t \geqq 0} .     

Three-term asymptotic expansions for the moments of the random walk with triangular distributed interference of chance

In this study, a semi-Markovian random walk with a discrete interference of chance (X(t)) is considered and under some weak assumptions the ergodicity of this process is discussed. The exact formulas for the first four moments of ergodic distribution of the process X(t) are obtained when the random variable ζ₁, which is describing a discrete interference of chance, has a triangular distribution in the interval [s, S] with center (S + s)/2. Based on these results, the asymptotic expansions with three-term are obtained for the first four moments of the ergodic distribution of X(t), as a ≡ (S − s)/2 → ∞. Furthermore, the asymptotic expansions for the variance, skewness and kurtosis of the ergodic distribution of the process X(t) are established. Finally, by using Monte Carlo experiments it is shown that the given approximating formulas provide high accuracy even for small values of parameter a.     

Exact Rates of Convergence of Functional Limit Theorems for Csörgő;-Révész Increments of a Wiener Process

Let {W(t); t≥ 0} be a standard Wiener process and S be the Strassen set of functions. We investigate the exact rates of convergence to zero (as T→∞) of the variables $ \sup _{{0 \leqslant t \leqslant T - \alpha _{T} }} \inf _{{f \in S}} \sup _{{0 \leqslant x \leqslant 1}} {\left| {Y_{{t,T}} {\left( x \right)} - f{\left( x \right)}} \right|} Let {W(t); t≥ 0} be a standard Wiener process and S be the Strassen set of functions. We investigate the exact rates of convergence to zero (as T→∞) of the variables sup_{0≤ t ≤ T − aT} inf _f∈S sup_{0≤ x ≤1}|Y _t,T (x) −f(x)| and inf_{0≤ t ≤ T−aT} sup_{0≤ x ≤1}|Y _t,T (x−f(x)| for any given f∈S, where Y _t,T (x) = (W(t+xa _T ) −W(t)) (2a _T (log Ta _T ⁻¹ + log log T))^−1/2. We establish a relation between how small the increments are and the functional limit results of Cs?rg{\H o}-Révész increments for a Wiener process. Similar results for partial sums of i.i.d. random variables are also given. Received September 10, 1999, Accepted June 1, 2000     

On a problem of Erdős concerning property B

A family ℱ of sets has propertyB if there exists a setS such thatS∩F≠0 andS⊄F for everyF∈ℱ. ℱ has propertyB(s) if there exists a setS such that 0<|F∩S|<s for everyF∈ℱ. Denote bym(n) (respectivelym(n, s)) the size of a smallest family ofn-element sets not having propertyB (respectivelyB(s)). P. Erdős has asked whetherm(n, s)≧m (s) for alln≧s. We show that, in general, this inequality does not hold.