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.     

Some divisibilities amongst the terms of linear recurrences

Let (u_n)n≥0 be a non-degenerate linear recurrence sequence of integers. We show that the set of positive integersn such that either ω)(n) orΩ(n) dividesu _n is of asymptotic density zero, where ω(n) and Ω(n) are the numbers of prime and prime power divisors ofn, respectively. The same also holds for the set of positive integersn such that τ(n)u _n , where τ(n) is the number of the positive integer divisors of n, provided thatu _n satisfies some mild technical conditions.     

On some properties of solutions of second-order linear functional differential equations

The properties of solutions of the equationu″(t) =p ₁(t)u(τ₁(t)) +p ₂(t)u′(τ₂(t)) are investigated wherep _i :a, + ∞[→R (i=1,2) are locally summable functions τ₁ :a, + ∞[→R is a measurable function, and τ₂ :a, + ∞[→R is a nondecreasing locally absolutely continuous function. Moreover, τ _i (t) ≥t (i = 1,2),p ₁(t)≥0,p ₂ ² (t) ≤ (4 - ɛ)τ ₂ ^′ (t)p ₁(t), ɛ =const > 0 and . In particular, it is proved that solutions whose derivatives are square integrable on [α,+∞] form a one-dimensional linear space and for any such solution to vanish at infinity it is necessary and sufficient that .     

Asymptotic normality and efficiency of a weighted correlogram

For a process X(t)=Σ _j=1 ^M g _j (t)ξ _j (), where g_j(t) are nonrandom given functions, is a stationary vector-valued Gaussian process, Eξ_k(t) = 0, and Eξ_k(0) Eξ_l(τ) = r _kl(τ), we construct an estimate for the functions r _kl(τ) on the basis of observations X(t), t ∈ [0, T]. We establish conditions for the asymptotic normality of as T → ∞. We consider the problem of the optimal choice of parameters of the estimate depending on observations. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 7, pp. 937–947, July, 1998.     

Real loci of based loop groups

Let (G, K) be a Riemannian symmetric pair of maximal rank, where G is a compact simply connected Lie group and K is the fixed point set of an involutive automorphism σ. This induces an involutive automorphism τ of the based loop space Ω(G). There exists a maximal torus T ⊂ G such that the canonical action of T × S ¹ on Ω(G) is compatible with τ (in the sense of Duistermaat). This allows us to formulate and prove a version of Duistermaat’s convexity theorem. Namely, the images of Ω(G) and Ω(G) ^τ (fixed point set of τ) under the T × S ¹ moment map on Ω(G) are equal. The space Ω(G) ^τ is homotopy equivalent to the loop space Ω(G/K) of the Riemannian symmetric space G/K. We prove a stronger form of a result of Bott and Samelson which relates the cohomology rings with coefficients in \mathbbZ₂ {\mathbb{Z}_2} of Ω(G) and Ω(G/K). Namely, the two cohomology rings are isomorphic, by a degree-halving isomorphism (Bott and Samelson [BS] had proved that the Betti numbers are equal). A version of this theorem involving equivariant cohomology is also proved. The proof uses the notion of conjugation space in the sense of Hausmann, Holm, and Puppe [HHP].     

Adjoint bi-continuous semigroups and semigroups on the space of measures

For a given bi-continuous semigroup (T(t)) _t⩾0 on a Banach space X we define its adjoint on an appropriate closed subspace X° of the norm dual X′. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology σ(X°,X). We give the following application: For Ω a Polish space we consider operator semigroups on the space C_b(Ω) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(Ω) of bounded Baire measures (endowed with the weak*-topology). We show that bi-continuous semigroups on M(Ω) are precisely those that are adjoints of bi-continuous semigroups on C_b(Ω). We also prove that the class of bi-continuous semigroups on C_b(ω) with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict topology. In general, if is not a Polish space this is not the case.     

Strong Arens irregularity of Beurling algebras with a locally convex topology

Let G be a locally compact group with a weight function ω. Recently, we have shown that the Banach space L₀^∞ (G,1/ω) can be identified with the strong dual of L¹(G, ω)equipped with some locally convex topologies τ. Here we use this duality to introduce an Arens multiplication on (L¹(G, ω), τ)**, and prove that the topological center of (L¹(G, ω), τ)** is (L¹(G, ω); this enables us to conclude that (L¹(G, ω), τ) is Arens regular if and only if G is discrete. We also give a characterization for Arens regularity of L₀^∞ (G, 1/ω)¹. Received: 8 March 2005     

On noncommutative weighted local ergodic theorems on <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-spaces

In the present paper we consider a von Neumann algebra M with a faithful normal semi-finite trace τ, and {α _t }, a strongly continuous extension to L ^p (M, τ) of a semigroup of absolute contractions on L ¹(M, τ). By means of a non-commutative Banach Principle we prove for a Besicovitch function b and x ∊ L ^p (M, τ), that the averages 1/T ∫₀ ^T b(t)α _t (x)dt converge bilateral almost uniformly in L ^p (M, τ) as T → 0. Communicated by Dénes Petz     

On representation theorem of <Emphasis Type="Italic">G</Emphasis>-expectations and paths of <Emphasis Type="Italic">G</Emphasis>-Brownian motion

We give a very simple and elementary proof of the existence of a weakly compact family of probability measures {Pθ ： θ∈θ} representing an important sublinear expectation- G-expectation E[·]. We also give a concrete approximation of a bounded continuous function X（ω） by an increasing sequence of cylinder functions Lip（Ω） in order to prove that Cb（Ω） belongs to the completion of Lip（Ω） under the natural norm E[｜·｜].     

Limiting process for integral functionals of a wiener process on a cylinder

We prove that integral functionals, whose integrands are bounded functions of a Wiener process on a cylinder, weakly converge to the processw ₁(τ(t)), τ(t) = β₁ t + (β₂ − β₁)mes {s:w ₂(s)≥0,s<t}, wherew ₁(t andw ₂(t) are independent one-dimensional Wiener processes, β₁ and β₂ are nonrandom values, and β₂≥β₁≥0. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 765–768, June, 1994.     

On eigenfunctions localized in a neighborhood of the lateral surface of a thin domain

We study the spectrum of the boundary-value problem for the Laplace operator in a thin domain Ω(ε) obtained by small perturbation of the cylinder Ω(ε)=ω×(-ε/2.ε/2) ⊂ ℝ³in a neighborhood of the lateral surface. The Dirichlet condition is imposed on the bases of the cylinder, and the Dirichlet condition or the Neumann condition is imposed on the remaining part of ∂Ω(ε). We construct and justify asymptotic formulas (as ε→+0) for eigenvalues and eigenfunctions. In view of a special form of the lateral surface, there are eigenfunctions of boundary-layer type that exponentially decrease far from the lateral surface. For the mixed boundary-value problem such a localization is possible in neighborhoods of local maxima of the curvature of the contour ∂ω. This property of eigenfunctions is a characteristic feature of the first points of the spectrum (in particular, the first eigenvalue) and, under the passage from Ω(h)₍₎ to Ω(h), the spectrum itself has perturbation O(h⁻²). Bibliography: 29 titles. Translated fromProblemy Matematicheskogo Analiza, No. 19, 1999, pp. 105–149.     

Estimates of the weak distance between finite-dimensional Banach spaces

We prove a variant of a theorem of N. Alon and V. D. Milman. Using it we construct for everyn-dimensional Banach spacesX andY a measure space Ω and two operator-valued functionsT: Ω→L(X, Y),S: Ω→L(Y, X) so that ∫_Ω S(ω)oT(ω)dω is the identity operator inX and ∫_Ω||S(ω)||·||T(ω)||dω=O(n ^α ) for some absolute constantα<1. We prove also that any subset of the unitn-cube which is convex, symmetric with respect to the origin and has a sufficiently large volume possesses a section of big dimension isomorphic to ak-cube. Research supported in part by a grant of the Israel Academy of Sciences.     

An Exceptional Set in the Ergodic Theory of Expanding Maps on Manifolds

Under a general hypothesis an expanding map T of a Riemannian manifold M is known to preserve a measure equivalent to the Liouville measure on that manifold. As a consequence of this and Birkhoff’s pointwise ergodic theorem, the orbits of almost all points on the manifold are asymptotically distributed with regard to this Liouville measure. Let T be Lipschitz of class τ for some τ in (0,1], let Ω(x) denote the forward orbit closure of x and for a positive real number δ and let E(x₀, δ) denote the set of points x in M such that the distance from x₀ to Ω is at least δ. Let dim A denote the Hausdorff dimension of the set A. In this paper we prove a result which implies that there is a constant C(T) > 0 such that dimE(x₀,d) 3 dimM - \fracC(T)|logd| \dim E(x_0,\delta) \ge \dim M - \frac{C(T)}{\vert\!\log \delta \vert} if τ = 1 and dimE(x₀,d) 3 dimM - \fracC(T)log|logd|\dim E(x_0,\delta) \ge \dim M - \frac{C(T)}{\log \vert \log \delta \vert} if τ < 1. This gives a quantitative converse to the above asymptotic distribution phenomenon. The result we prove is of sufficient generality that a similar result for expanding hyperbolic rational maps of degree not less than two follows as a special case.     

Maximal subspaces for solutions of the second order abstract cauchy problem

For a continuous, increasing function ω: R → R \{0} of finite exponential type, this paper introduces the set Z(A, ω) of all x in a Banach space X for which the second order abstract differential equation (2) has a mild solution such that [ω(t)]-1u(t,x) is uniformly continues on R , and show that Z(A, ω) is a maximal Banach subspace continuously embedded in X, where A ∈ B(X) is closed. Moreover, A|z(A,ω) generates an O(ω(t))strongly continuous cosine operator function family.     

Critical exponent for a quasilinear parabolic equation with inhomogeneous density in a cone

In this paper, we study the initial-boundary value problem of porous medium equation ρ(x)u _t  = Δu ^m  + V(x)h(t)u ^p in a cone D = (0, ∞) × Ω, where V(x)  ~  |x|^s, h(t)  ~  t^s{V(x)\,{\sim}\, |x|^\sigma, h(t)\,{\sim}\, t^s}. Let ω ₁ denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on Ω and let l denote the positive root of l ² + (n − 2)l = ω ₁. We prove that if m < p ￡ 1+(m-1)(1+s)+\frac2(s+1)+sn+l{m < p \leq 1+(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}, then the problem has no global nonnegative solutions for any nonnegative u ₀ unless u ₀ = 0; if ${p >1 +(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}${p >1 +(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}, then the problem has global solutions for some u ₀ ≥ 0.     

Multiscale stochastic homogenization of convection-diffusion equations

Multiscale stochastic homogenization is studied for convection-diffusion problems. More specifically, we consider the asymptotic behaviour of a sequence of realizations of the form ∂u _ɛ ^ω / ∂t+1 / ɛ ₃ C(T ₃(x/ɛ ₃)ω ₃) · ∇u _ɛ ^ω − div(α(T ₂(x/ɛ ₂)ω ₂, t) ∇u _ɛ ^ω ) = f. It is shown, under certain structure assumptions on the random vector field C(ω ₃) and the random map α(ω ₁, ω ₂, t), that the sequence {u _ɛ ^ω } of solutions converges in the sense of G-convergence of parabolic operators to the solution u of the homogenized problem ∂u/∂t − div (B(t)∇u= f).     

Convergence to equilibria for compressible navier-stokes equations with large data

We show that for any sequence t_n→∞ and any global weak solution (ρ(t), u(t)) of barotropic compressible Navier-Stokes equations in 2 and 3 space dimensions with potential force there exists a subsequence {s_n} such that ϱ(s_n) in L _ω ^r (Ω) of spatially periodic functions, where ϱ_∞ is an equilibrium density, r>1 suitable. If the equilibrium is unique then the convergence holds for all t→∞. No smallness of data is assumed. Entrata in Redazione il 22 febbraio 1999. Ricevuta nuova versione il 24 agosto 1999. This work was started and completed during the stay of the second author at the University of Toulon in March 1997 and April 1998, which is acknowledged for the financial support, and also partially supported by the Grant Agency of the Czech Republic, Grant No. 201/96/0313.     

Estimates of the Bergman kernel for some pseudoconvex domains

Denote by K_ω(z, ζ) the Bergman kernel of a pseudoconvex domain Ω. For some classes of domains Ω, a relationship is found between the rate of increase of K_ω(z, z) as z tends to ∂Ω, and a purely geometric property of Ω. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 222, 1995, pp. 222–245.     

A preferential attachment model with random initial degrees

In this paper, a random graph process {G(t)} _t≥1 is studied and its degree sequence is analyzed. Let {W _t } _t≥1 be an i.i.d. sequence. The graph process is defined so that, at each integer time t, a new vertex with W _t edges attached to it, is added to the graph. The new edges added at time t are then preferentially connected to older vertices, i.e., conditionally on G(t-1), the probability that a given edge of vertex t is connected to vertex i is proportional to d _i (t-1)+δ, where d _i (t-1) is the degree of vertex i at time t-1, independently of the other edges. The main result is that the asymptotical degree sequence for this process is a power law with exponent τ=min{τ_W,τ_P}, where τ_W is the power-law exponent of the initial degrees {W _t } _t≥1 and τ_P the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze.     

Asymptotic representations of solutions of nonautonomous ordinary differential equations with regularly varying nonlinearities

We obtain asymptotic representations as t ↑ ω, ω ≤ + ∞, for all possible types of P _ω(Y ₀, λ ₀)-solutions (where Y ₀ is zero or ±∞ and −∞ ≤ λ₀ ≤ +∞) of nonlinear differential equations y ⁽ⁿ⁾ = α ₀ p(t)φ(y), where α ₀ ∈ {−1, 1}, p: [a, ω[→]0,+∞[ is a continuous function, and φ is a continuous regularly varying function in a one-sided neighborhood of Y ₀.