Convolution products of probability measures on completely simple semigroups

Convolution products of probability measures are considered in the context of completely simple semigroups. Given a sequence of measures (μn)⊂Prob(S), where S is a finite completely simple semigroup, results are proven which (1) relate the supports of the measures in the sequence to the supports of the tail limit measures, and (2) determine necessary and sufficient conditions for convergence of the convolution products in the case of rectangular groups. An example showing how the theory can be used to analyze the convergence behavior of non-homogeneous Markov chains is included.     

Tail behavior for semigroups of measures on groups

Let (_t)_t>0 be a convolution semigroup of probability measures on a measurable group (G, ). In this paper, we provide precise information about the asymptotic behavior of _t{q>s, whereq is a measurable seminorm and (_t)_t>0 isq-continuous.     

Maximal Homomorphic Group Image and Convergence of Convolution Sequences on a Semigroup

Let be a probability measure generating a locally compact semigroup S. If the convolution sequence ⁿ is tight, in particular if S is compact, S admits a closed minimal ideal K. The convergence of ⁿ is characterized in terms of convergence of a homomorphic image (~) ⁿ on a factor group of the compact group G in the Rees–Suschkewitsch decomposition of K.     

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.     

Harris-type results on geometric and subgeometric convergence to equilibrium for stochastic semigroups

We provide simple and constructive proofs of Harris-type theorems on the existence and uniqueness of an equilibrium and the speed of equilibration of discrete-time and continuous-time stochastic semigroups. Our results apply both to cases where the relaxation speed is exponential (also called geometric) and to those with no spectral gap, with non-exponential speeds (also called subgeometric). We give constructive estimates in the subgeometric case and discrete-time statements which seem both to be new. The method of proof also differs from previous works, based on semigroup and interpolation arguments, valid for both geometric and subgeometric cases with essentially the same ideas. In particular, we present very simple new proofs of the geometric case.     

Optimal convergence rate for Yosida approximations of bounded holomorphic semigroups

In this paper, we obtain optimal bounds for convergence rate for Yosida approximations of bounded holomorphic semigroups. We also provide asymptotic expansions for semigroups in terms of Yosida approximations and obtain optimal error bounds for these expansions.     

Necessary and sufficient conditions for conservativeness of dynamical semigroups

Dynamical semigroups constitute a quantum-mechanical generalization of Markov semigroups, a concept familiar from the theory of stochastic processes. Let be a Hilbert space andA a von Neumann algebra. A dynamical semigroup P_t is a -weakly continuous one-parameter semigroup of completely positive maps ofA into itself. A semigroup P_t possessing the property of preserving the identityIA is said to be conservative and its infinitesimal operator L[·] is said to be regular. The present paper studies necessary and sufficient conditions for strongly continuous dynamical semigroups to be conservative. It is shown that under certain additional assumptions one can formulate necessary and sufficient conditions which are analogous to Feller's condition for regularity of a diffusion process: the equation P=L[P] has no solutions inA ₊. Using a Jensen-type inequality for completely positive maps, constructive sufficient conditions are obtained for conservativeness, in the form of inequalities for commutators. The restriction of a dynamical subgroup to an Abelian subalgebra of (R ⁿ ) yields a series of new regularity conditions for both diffusion and jump processes.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 36, pp. 149–184, 1990.     

Necessary and sufficient conditions for the convergence of iterative methods for the linear complementarity problem

Necessary and sufficient conditions are established for the convergence of various iterative methods for solving the linear complementarity problem. The fundamental tool used is the classical notion of matrix splitting in numerical analysis. The results derived are similar to some well-known theorems on the convergence of iterative methods for square systems of linear equations. An application of the results to a strictly convex quadratic program is also given.This research was based on work supported by the National Science Foundation under Grant No. ECS-81-14571.The author gratefully acknowledges several comments by K. Truemper on the topics of this paper.     

Some Results Concerning Convergence of Convolution Products of Probability Measures on Discrete Semigroups

Two types of conditions have been significant when considering the convergence of convolution products of nonidentical probability measures on groups and semigroups. The essential points of a sequence of measures have been useful in characterizing the supports of the limit measures. Also, enough mass eventually on an idempotent has proven sufficient for convergence in a number of structures. In this paper, both of these types of conditions are analyzed in the context of discrete non-abelian semigroups. In addition, an application to the convergence of nonhomogeneous Markov chains is given.     

Necessary and sufficient conditions for convergence of attainment times

Translated fromProblemy Ustoichivosti Stokhasticheskikh Modelei. Trudy Seminara, 1988, pp. 129–137.     

Necessary and sufficient conditions for the convergence to nonquasicontinuous semimartingales

This paper was prepared for the conference on stochastic differential systems held in Baku in 1984.     

On weak convergence of random fields

We provide a characterization of compactness in the spaceD of functions of two variables defined on a unit square. The functions fromD have the property that their discontinuity points lie on smooth curves. Conditions for the tightness of probability measures inD and conditions for weak convergence of random fields with trajectories inD are derived. Vilnius Gediminas Technical University, Saulétekio 11; Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 2, pp 169–184, April–June, 1999. Translated by R. Banys     

Necessary and sufficient well-posedness conditions for a convolution equation of the second kind with even kernel on a finite interval

We derive some necessary and sufficient conditions for the well-posedness of a convolution equation of the second kind with even kernel on a finite interval. In order to check these conditions it suffices to compute a one-dimensional integral (of a given function) with precision less than 0.5. As an intermediate result we give a strengthening of the Fredholm alternative for the equation in question with an arbitrary kernel in L ₁.     

Strong convergence of modified Ishikawa iteration for two asymptotically nonexpansive mappings and semigroups

In this paper, we established strong convergence theorems for a common fixed point of two asymptotically nonexpansive mappings and for a common fixed point of two asymptotically nonexpansive semigroups by using the hybrid method in a Hilbert space. Moreover, we also proved a strong convergence theorem for a common fixed point of two nonexpansive mappings. Our results extend and improve the recent ones announced by Kim and Xu [T.W. Kim, H.W. Xu, Strong convergence of modified Mann iteration for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal. 64 (2006) 1140–1152], Nakajo and Takahashi [K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379], and many others.     

Poincaré and Opial inequalities for vector-valued convolution products

Various L^p$L^p$ form Poincaré and Opial inequalities are given for vector-valued convolution products. We apply our results to infinitesimal generators of _C0$C_0$-semigroups and cosine functions. Typical examples of these operators are differential operators in Lebesgue spaces.     

Necessary and sufficient conditions for the convergence of two- and three-point Newton-type iterations

Necessary and sufficient conditions under which two- and three-point iterative methods have the order of convergence р (2 ≤ р ≤ 8) are formulated for the first time. These conditions can be effectively used to prove the convergence of iterative methods. In particular, the order of convergence of some known optimal methods is verified using the proposed sufficient convergence tests. The optimal set of parameters making it possible to increase the order of convergence is found. It is shown that the parameters of the known iterative methods with the optimal order of convergence have the same asymptotic behavior. The simplicity of choosing the parameters of the proposed methods is an advantage over the other known methods.     

Strong continuity of generalized Feynman-Kac semigroups: Necessary and sufficient conditions

Let (E,D(E)) be a strongly local, quasi-regular symmetric Dirichlet form on L²(E;m) and ((Xt)_t?0,(Px)_x∈E) the diffusion process associated with (E,D(E)). For u∈De(E), u has a quasi-continuous version and has Fukushima's decomposition: , where is the martingale part and is the zero energy part. In this paper, we study the strong continuity of the generalized Feynman-Kac semigroup defined by , t?0. Two necessary and sufficient conditions for to be strongly continuous are obtained by considering the quadratic form (Qu,Db(E)), where Qu(f,f):=E(f,f)+E(u,f²) for f∈Db(E), and the energy measure μ〈u〉 of u, respectively. An example is also given to show that is strongly continuous when μ〈u〉 is not a measure of the Kato class but of the Hardy class with the constant (cf. Definition 4.5).     

Necessary and sufficient conditions for the existence of exponential attractors for semigroups,and applications

First we establish some necessary and sufficient conditions for the existence of exponential attractors by using ω$\omega$-limit compactness and a measure of non-compactness. Then we provide a new method for proving the existence of exponential attractors. We prove the existence of exponential attractors for reaction–diffusion equations and 2D Navier–Stokes equations as simple applications.