Conservative markov processes on a topological space

A Markov operator preservingC(X) is known to induce a decomposition of the locally compact spaceX to conservative and dissipative parts. Two notions of ergodicity are defined and the existence of subprocesses is studied. A sufficient condition for the existence of a conservative subprocess is given, and then the process is assumed to be conservative. When it has no subprocesses, sufficient conditions for the existence of a σ-finite invariant measure are given, and are extended to continuous-time processes. When the invariant measure is unique, ratio limit theorems are proved for the discrete and continuous time processes. Examples show that some combinations of conservative processes are not necessarily conservative. This paper is a part of the authors’s Ph.D. thesis prepared at the Hebrew University under the direction of Professor S. R. Foguel, to whom the author is grateful for his helpful advice and kind encouragement.     

A Banach space in which all compact sets, but not all bounded sets, admit Chebyshev centers

Given a Banach spaceX, letc ₀(X) be the space of all null sequences inX (equipped with the supremum norm). We show that: 1) each compact set inc ₀(X) admits a (Chebyshev) center iff each compact set inX admits a center; 2) forX satisfying a certain condition (Q), each bounded set inc ₀(X) admits a center iffX is quasi uniformly rotund. We construct a Banach spaceX such that the compact subsets ofX admit centers,X satisfies the condition (Q) andX is not quasi uniformly rotund. It follows that the Banach spaceE=c ₀(X) has the property from the title. Eine überarbeitete Fassung ging am 4. 7. 2001 ein     

Non-baire unions in category bases

We consider a partition of a spaceX consisting of a meager subset ofX and obtain a sufficient condition for the existence of a subfamily of this partition which gives a non-Baire subset ofX. The condition is formulated in terms of the theory of J. Morgan [1].     

On the existence of invariant probability measures

Let (X,A) be a measureable space andT:X →X a measurable mapping. Consider a family ℳ of probability measures onA which satisfies certain closure conditions. IfA ₀⊂A is a convergence class for ℳ such that, for everyA ∈A ₀, the sequence ((1/n) Σ _i ^=0/n−1 1 _A ∘T ⁱ) converges in distribution (with respect to some probability measurev ∈ ℳ), then there exists aT-invariant element in ℳ. In particular, for the special case of a topological spaceX and a continuous mappingT, sufficient conditions for the existence ofT-invariant Borel probability measures with additional regularity properties are obtained.     

Qualitative behaviour of stochastic delay equations with a bounded memory

A dichotomy is proved concerning recurrence properties of the solution of certain stochastic delay equations. If the solution process is recurrent, there exists an invariant measure π on the state space C which is unique (up to a multiplicative constant) and the tail-field is trivial. If π happens to be a probability measure, then for every initial condition, the distribution of the process converges to it as t→∞. We will formulate a sufficient condition for the existence of an invariant probability measure (ipm) in icrnia of Lyapunov junctionals and give two examples, one Heing the stochastic-delay version of the famous logistic equation of population growth. Finally we study approximations of delay equations by Markov chains.     

Foliations of polynomial growth are hyperfinite

We define a class of equivalence relations with polynomial growth and show that such relations always support finite invariant measures and are hyperfinite. In particular, foliations of polynomial growth define hyperfinite equivalence relations with respect to any family of finite invariant measures on transversals. We also extend a result of Dye for countable groups to show that if a locally compact second countable groupG acts freely on a Lebesgue spaceX with finite invariant measure, so that the orbit relation onX is hyperfinite, thenG is amenable.     

Topological direct sum decompositions of banach spaces

LetY andZ be two closed subspaces of a Banach spaceX such thatY≠lcub;0rcub; andY+Z=X. Then, ifZ is weakly countably determined, there exists a continuous projectionT inX such that ∥T∥=1,T(X)⊃Y, T ⁻¹(0)⊂Z and densT(X)=densY. It follows that every Banach spaceX is the topological direct sum of two subspacesX ₁ andX ₂ such thatX ₁ is reflexive and densX ₂**=densX**/X.     

The space of maps from a locally compact space to a Banach space

The space of continuous maps from a topological spaceX to topological spaceY is denoted byC(X,Y) with the compact-open topology. In this paper we prove thatC(X,Y) is an absolute retract ifX is a locally compact separable metric space andY a convex set in a Banach space. From the above fact we know thatC(X,Y) is homomorphic to Hilbert spacel ₂ ifX is a locally compact separable metric space andY a separable Banach space; in particular,C(R ⁿ,R^m) is homomorphic to Hilbert spacel ₂. This research is supported by the Science Foundation of Shanxi Province's Scientific Committee     

InvariantR-lineals of some continuous additive operators

We prove that an infinite-dimensional Banach spaceX contains a nontrivial closedR-linealX ₀ invariant both under the action of a compact additive operatorA and under the action of all continuous additive operatorsT inX suchthatT=T ₁+T ₂, whereT ₁ A=AT ₁ andT ₂ A=–AT ₂. IfA is a linear or antilinear compact operator, thenX ₀ is a subspace ofX.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 5, pp. 611–615, May, 1995.     

Renorming in the space of Bochner integrable functionsL 1(μ,X)

A sufficient condition is given when a subspaceL⊂L ₁(μ,X) of the space of Bochner integrable function, defined on a finite and positive measure space (S, Φ, μ) with values in a Banach spaceX, is locally uniformly convex renormable in terms of the integrable evaluations {∫ _A fdμ;f∈L}. This shows the lifting property thatL ₁(μ,X) is renormable if and only ifX is, and indicates a large class of renormable subspaces even ifX does not admit and equivalent locally uniformly convex norm.     

A simple characterization of the set ofμ-entropy pairs and applications

We present simple characterizations of the setsE _μ andE _X of measure entropy pairs and topological entropy pairs of a topological dynamical system (X, T) with invariant probability measureμ. This characterization is used to show that the set of (measure) entropy pairs of a product system coincides with the product of the sets of (measure) entropy pairs of the component systems; in particular it follows that the product of u.p.e. systems (topological K-systems) is also u.p.e. Another application is to show that the proximal relationP forms a residual subset of the setE _X . Finally an example of a minimal point distal dynamical system is constructed for whichE _X ∩(X ₀×X ₀)≠ , whereX ₀ is the denseG _δ subset of distal points inX.     

On the existence of smooth densities for jump processes

Summary We consider a Lévy processX _t and the solutionY _t of a stochastic differential equation driven byX _t; we suppose thatX _t has infinitely many small jumps, but its Lévy measure may be very singular (for instance it may have a countable support). We obtain sufficient conditions ensuring the existence of a smooth density forY _t: these conditions are similar to those of the classical Malliavin calculus for continuous diffusions. More generally, we study the smoothness of the law of variablesF defined on a Poisson probability space; the basic tool is a duality formula from which we estimate the characteristic function ofF.     

On the existence ofσ-finite invariant measures for operators

Several necessary and sufficient conditions are given for the existence of aσ-finite invariant measure for a positive operator onL _∞. They are ofσ-type: the entire space is an increasing union of setsX _k each of which is well-behaved. To the Memory of Shlomo Horowitz Research in part supported by the National Science Foundation (U.S.A.).     

Densely two-valued metric projections in uniformly convex Banach spaces

For every uniformly convex Banach spaceX with dimX2 there is a residual setU in the Hausdorff metric spaceB(X) of bounded and closed sets inX such that the metric projection generated by a set fromU is two-valued and upper semicontinuous on a dense and everywhere continual subset ofX. For any two closed and separated subsetsM ₁ andM ₂ ofX the points on the equidistant hypersurface which have best approximations both inM ₁ andM ₂ form a dense G set in the induced topology.The author is partially supported by the National Fund for Scientific Research at the Bulgarian Ministry of Science and Education under contract MM 408/94.     

Frobenius-Perron operators and approximation of invariant measures for set-valued dynamical systems

A set-valued dynamical systemF on a Borel spaceX induces a set-valued operatorF onM(X) — the set of probability measures onX. We define arepresentation ofF, each of which induces an explicitly defined selection ofF; and use this to extend the notions of invariant measure and Frobenius-Perron operators to set-valued maps. We also extend a method ofS. Ulam to Markov finite approximations of invariant measures to the set-valued case and show how this leads to the approximation ofT-invariant measures for transformations , whereT corresponds to the closure of the graph of .     

Function spaces not containing ℓ<Subscript>1</Subscript>

For Ω bounded and open subset of andX a reflexive Banach space with 1-symmetric basis, the function spaceJF _X (Ω) is defined. This class of spaces includes the classical James function space. Every member of this class is separable and has non-separable dual. We provide a proof of topological nature thatJF _X (Ω) does not contain an isomorphic copy of ℓ₁. We also investigate the structure of these spaces and their duals.     

The problem of envelopes for Banach spaces

LetX be a Banach space. A Banach spaceY is an envelope ofX if (1)Y is finitely representable inX; (2) any Banach spaceZ finitely representable inX and of density character not exceeding that ofY is isometric to a subspace ofY. Lindenstrauss and Pelczynski have asked whether any separable Banach space has a separable envelope. We give a negative answer to this question by showing the existence of a Banach space isomorphic tol ₂, which has no separable envelope. A weaker positive result holds: any separable Banach space has an envelope of density character ≦ℵ₁ (assuming the continuum hypothesis).     

The existence of invariant probability measures for a group of transformations

SupposeG is a group of measurable transformations of aσ-finite measure space (X,A, m). A setA ∈A is weakly wandering underG if there are elementsg _n ∈G such that the setsg _nA, n=0, 1,…, are pairwise disjoint. We prove that the non-existence of any set of positive measure which is weakly wandering underG is a necessary and sufficient condition for the existence of aG-invariant, probability measure defined onA and dominating the measurem in the sense of absolute continuity. This paper was written while the author was visiting the Technische Universitat Berlin as a research fellow of the Alexander von Humboldt Foundation.     

Multiplicative perturbations in semigroup theory with the (Z)-condition

In [4] the (Z)-condition was introduced by G. W. Desch and W. Schappacher. This condition on a Banach spaceZ and a generatorA inX ensures thatA(I+B) and(I+B)A generateC ₀-semigroups, ifB has its range inZ. In this paper we will consider how certain properties of the semigroup generated byA are inherited by the semigroups generated byA(I+B) and(I+B)A. We shall furthermore investigate a related condition, that simplifies certain sufficiency assumptions given previously.