Convolution semigroups and central limit theorem associated with a dual convolution structure

We consider hypergroups associated with Jacobi functions ⁽⁾ (x), (–1/2). We prove the existence of a dual convolution structure on [0,+[i(]0,s ₀]{{) =++1,s ₀=min(,–+1). Next we establish a Lévy-Khintchine type formula which permits to characterize the semigroup and the infinitely divisible probabilities associated with this dual convolution, finally we prove a central limit theorem.     

Harmonic functions on hypergroups

We initiate a study of harmonic functions on hypergroups. In particular, we introduce the concept of a nilpotent hypergroup and show such hypergroup admits an invariant measure as well as a Liouville theorem for bounded harmonic functions. Further, positive harmonic functions on nilpotent hypergroups are shown to be integrals of exponential functions. For arbitrary hypergroups, we derive a Harnack inequality for positive harmonic functions and prove a Liouville theorem for compact hypergroups. We discuss an application to harmonic spherical functions.     

Normed Convergence Property for Hypergroups Admitting An Invariant Measure

We prove that a hypergroup admitting a countable basis and an invariant Haar measure has normed convergence property if and only if it is compact.AMS Subject Classification (2001): 43A62, 60B99     

Hypercontractivity and Absolute Continuity of Gaussian Generalized Mehler Semigroups in Hilbert Spaces

In this paper, we prove the hypercontra,ctivity of a non-differentiable Gaussian generalized Mehler semigroup using direct probabilistic argumcents, This result implics the exponential convergence of the scmigroup at infinity. Under some additional hypotheses, we also） establish the absolute continuity of the semigroup with respect to its invariant mcasure.     

Crossed product of C*‐algebras by hypergroups

The crossed product of ‐algebras by groups, groupoids and semigroups are well studied. In this paper we introduce and study the crossed product of ‐algebras by (locally compact) hypergroups. We calculate the crossed products by finite hypergroups of orders 2 and 3.     

一类紧半群上概率测度卷积幂的弱收敛性

本文讨论紧半群上概率测度卷积幂的弱收敛性，将紧群上的Ｋａｗａｄａ－Ｉｔ６型结果以相应的形式建立到一类紧半群上．本文的结论蕴含了［１］中的定理２．１．４与［２］中的定理１．     

On the factor sets of measures and local tightness of convolution semigroups over Lie groups

It is shown that for a large class of Lie groups (called weakly algebraic groups) including all connected semisimple Lie groups the following holds: for any probability measure on the Lie group the set of all two-sided convolution factors is compact if and only if the centralizer of the support of inG is compact. This is applied to prove that for any connected Lie groupG, any homomorphism of any real directed (submonogeneous) semigroup into the topological semigroup of all probability measures onG is locally tight.     

On the theorem on asymptotic equidistribution of the convolution powers of symmetric measures on a unimodular group

The theorem on the asymptotic equidistribution of the convolution powers of a symmetric probability measure given on a unimodular group [1] deals with a measure whose convolution powers, starting from one of them, load an arbitrary prescribed nonempty subset of the group. In the present note, we indicate the modifications that arise under the replacement of the above condition by the requirement that the smallest closed subgroup (of the group considered) generated by the support of this measure coincide with the group itself. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 120–126, July, 1996. This research was partially supported by the International Science Foundation under grant No. 3J000.     

局部紧H半群上概率测度卷积幂的弱收敛性

本文讨论局部紧Ｈ半概率测度卷积幂的弱收敛性,将紧群上的Ｋａｗａｄａ－Ｉｔｏ型结果以某种相应的形式建立到局部紧Ｈ半群,由于紧半群上的概率测度卷积幂序列必为 ,所以,不仅Ｋａｗａｄａ－Ｉｔｏ经典结果是本文的特例,而且「１」中的定理和「２」中的定理２都可以作为本文定理的推论     

Automorphisms on a Lie group contracting modulo a compact subgroup and applications to semistable convolution semigroups

Let ( _t ) _t0 be a -semistable convolution semigroup of probability measures on a Lie groupG whose idempotent ₀ is the Haar measure on some compact subgroupK. Then all the measures ₁ are supported by theK-contraction groupC _K() of the topological automorphism ofG. We prove here the structure theoremC _K()=C()K, whereC() is the contraction group of . Then it turns out that it is sufficient to study semistable convolution semigroups on simply connected nilpotent Lie groups that have Lie algebras with a positive graduation.     

Convolution products of nonidentical distributions on a topological semigroup

In this paper we present a number of sufficient conditions on a sequence of probability measuresµ _n on a locally compact (second countable) Hausdorff topological semigroupS that guarantee the weak convergence of the sequence of convolution productsµ _k,n µ _{k + 1} *···*µ _n (k), asn, for allk0.     

Perturbation theory for convolution semigroups

We deal with convolution semigroups (not necessarily symmetric) in Lp(RN) and provide a general perturbation theory of their generators by indefinite singular potentials. Such semigroups arise in the theory of Lévy processes and cover many examples such as Gaussian semigroups, α-stable semigroups, relativistic Schrödinger semigroups, etc. We give new generation theorems and Feynman-Kac formulas. In particular, by using weak compactness methods in L¹, we enlarge the extended Kato class potentials used in the theory of Markov processes. In L² setting, Dirichlet form-perturbation theory is finely related to L¹-theory and the extended Kato class measures is also enlarged. Finally, various perturbation problems for subordinate semigroups are considered.     

Continuity and generators of dynamical semigroups for infinite Bose systems

For a class of quasifree quantum dynamical semigroups on the algebra of the canonical commutation relations (CCR) we give sufficient conditions for these semigroups to extend to ultraweakly continuous semigroups of normal operators on the von Neumann algebra associated with a representation of the CCR. Then the explicit form of the generators of the extended semigroups is calculated.     

Method of moments on semigroups

If ( _j ) is a sequence of measures onR ^k having momentss _n ( _j ) of all ordersnN ₀ ^k and if for eachnN ₀ ^k the sequence (s _{n j} )) _jN converges to somet _n R then some subsequence of ( _j ) converges weakly to a measure with moments of all orders satisfyings _n ()=t _n for allnN^0/k . Thisindeterminate method of moments and the continuity theorems in probability theory suggest a common generalization, dealing with a commutative semigroupS, with involution and a neutral element, and measures on the dual semigroupS ^* ofcharacters on S—hermitian multiplicative complex functions not identically zero. In this setting, a continuity theorem holds for measures on the set of bounded characters,⁽²⁾ and an indeterminate method of moments whenS is finitely generated.⁽²⁾ The latter result is generalized in the present paper to the case of arbitraryS. This leads to a generalization of Haviland's criterion for theK-moment problem, and to a continuity theorem for the so-called perfect semigroups.     

Continuity properties of decomposable probability measures on euclidean spaces

It is shown that every full e^A decomposable probability measure on R^k, where A is a linear operator all of whose eigenvalues have negative real part, is either absolutely continuous with respect to Lebesgue measure or continuous singular with respect to Lebesgue measure. This result is used to characterize the continuity properties of random variables that are limits of solutions of certain stochastic difference equations.     

Semigroup closures of finite rank symmetric inverse semigroups

We introduce the notion of semigroup with a tight ideal series and investigate their closures in semitopological semigroups, particularly inverse semigroups with continuous inversion. As a corollary we show that the symmetric inverse semigroup of finite transformations I _λ ⁿ of the rank ≤ n is algebraically closed in the class of (semi)topological inverse semigroups with continuous inversion. We also derive related results about the nonexistence of (partial) compactifications of classes of semigroups that we consider.     

Fixed point properties of semigroups of non-expansive mappings

In recent years, there have been considerable interests in the study of when a closed convex subset K of a Banach space has the fixed point property, i.e. whenever T is a non-expansive mapping from K into K, then K contains a fixed point for T. In this paper we shall study fixed point properties of semigroups of non-expansive mappings on weakly compact convex subsets of a Banach space (or, more generally, a locally convex space). By considering the classes of bicyclic semigroups we answer two open questions, one posted earlier by the first author in 1976 (Dalhousie) and the other posted by T. Mitchell in 1984 (Virginia). We also provide a characterization for the existence of a left invariant mean on the space of weakly almost periodic functions on separable semitopological semigroups in terms of fixed point property for non-expansive mappings related to another open problem raised by the first author in 1976.     

Density and tails of unimodal convolution semigroups

We give sharp bounds for the isotropic unimodal probability convolution semigroups when their Lévy–Khintchine exponent has Matuszewska indices strictly between 0 and 2.     

关于乘积Borel概率测度的一点注记

In this paper,a necessary and sufficient condition of a measure to be the product Borel probability measure on the product space of some compact metric spaces are given.     

Necessary and sufficient conditions for the convergence of convolution products of non-identical distributions on finite abelian semigroups

Given a sequence of probability measures ( _n ) on a finite abelian semigroup, we present necessary and sufficient conditions which guarantee the weak convergence of the convolution products _{k,n k+1}*···* _n (k<n), asn for allk0. These conditions are verifiable in the sense that they are based entirely on the individual measures in the sequence ( _n ).