Multiple Ergodicity For Reversible Markov Chains

Inspired by the multiple recurrence and multiple ergodic theorems for measure preserving systems, we discuss an analogous question for measure preserving semigroups. In this note, we deal with the symmetric semigroups associated to reversible Markov chains.     

保等价部分变换半群的变种半群上的正则元

在现有的保等价部分变换半群的基础上,引入了一个新的运算,得出保等价部分变换半群的变种半群的概念,利用格林关系及幂等元的正则性,讨论了这类半群中元素的正则性,给出了保等价部分变换半群的变种半群中一个元是正则元的充要条件     

On a class of abstract neutral functional differential equations

By using the theory of semigroups of growth α, we discuss the existence of mild solutions for a class of abstract neutral functional differential equations. A concrete application to partial neutral functional differential equations is considered.     

Semigroups of locally Lipschitz operators associated with semilinear evolution equations

In this paper we introduce the notion of semigroups of locally Lipschitz operators which provide us with mild solutions to the Cauchy problem for semilinear evolution equations, and characterize such semigroups of locally Lipschitz operators. This notion of the semigroups is derived from the well-posedness concept of the initial-boundary value problem for differential equations whose solution operators are not quasi-contractive even in a local sense but locally Lipschitz continuous with respect to their initial data. The result obtained is applied to the initial-boundary value problem for the complex Ginzburg–Landau equation.     

Intertwined evolution operators

We construct a convolution algebra of admissible homomorphisms defined on a ‘test space’ to demonstrate the fundamental role of convolution in the study of intertwined evolution operators of linear ordinary differential equations in Banach spaces and probability theory. The choice of test space makes the framework we present quite versatile. The applications include semigroups of linear operators, empathy, integrated semigroups and empathies and the convolution semigroups of probability theory.     

A Hille-Yosida theorem for weakly continuous semigroups

We introduce a new class of weakly continuous semigroups and give a characterization of their infinitesimal generators, generalizing the classical Hille-Yosida Theorem for strongly continuous semigroups. The results are illustrated by the example of transition semigroups corresponding to the solutions of certain stochastic differential equations.     

抽象半线性发展方程初值问题的整体解

在抽象空间框架下,研究了具有广泛物理背景的一类半线性发展方程初值问题整体解的存在性。利用正算子半群特征与凸锥理论,把上下解方法引入该问题,给出了整体解存在及唯一的若干充分条件。所得的结果概括、统一及推广了常微分方程、偏微分方程及Banach空间常微分方程中的有关结论。     

Long time behavior of solutions to semilinear parabolic equations involving strongly degenerate elliptic differential operators

In this paper we consider initial boundary value problem for semilinear parabolic equations involving strongly degenerate elliptic differential operators. Depending on the concrete types of nonlinearity we establish the existence of compact connected global attractors of semigroups generated by the problem under consideration.     

Asymptotic behavior of one-parameter semigroups of positive operators

In this paper we survey the Perron-Frobenius spectral theory for positive semigroups on Banach lattices and indicate its applications to stability theory of retarded differential equations and quasi-periodic flows.     

A Characteristic Equation for Non-autonomous Partial Functional Differential Equations

We characterize the exponential dichotomy of non-autonomous partial functional differential equations by means of a spectral condition extending known characteristic equations for the autonomous or time-periodic case. From this we deduce robustness results. We further study the almost periodicity of solutions to the inhomogeneous equation. Our approach is based on the spectral theory of evolution semigroups.     

On a class of exponential-type operators and their limit semigroups

The paper is mainly focused upon the study of a class of second order degenerate elliptic operators on unbounded intervals.We show that these operators generate strongly continuous semigroups in suitable weighted spaces of continuous functions.Furthermore, we represent the semigroups as limits of iterates of the so-called exponential-type operators.In a particular case, starting from the stochastic differential equations associated with these operators, we also find an integral representation of the semigroup and determine its asymptotic behaviour.     

Extremal equilibria for monotone semigroups in ordered spaces with application to evolutionary equations

We consider monotone semigroups in ordered spaces and give general results concerning the existence of extremal equilibria and global attractors. We then show some applications of the abstract scheme to various evolutionary problems, from ODEs and retarded functional differential equations to parabolic and hyperbolic PDEs. In particular, we exhibit the dynamical properties of semigroups defined by semilinear parabolic equations in RN with nonlinearities depending on the gradient of the solution. We consider as well systems of reaction-diffusion equations in RN and provide some results concerning extremal equilibria of the semigroups corresponding to damped wave problems in bounded domains or in RN. We further discuss some nonlocal and quasilinear problems, as well as the fourth order Cahn-Hilliard equation.     

Special functions as subordinated semigroups on the real line

We give examples of convolution semigroups on the positive half-line and on the real line. Such semigroups are expressed in terms of special functions which arise in classical differential equations.     

A comparison of numerical integration rules for the meshless local Petrov–Galerkin method

The meshless local Petrov–Galerkin (MLPG) method is a mesh-free procedure for solving partial differential equations. However, the benefit in avoiding the mesh construction and refinement is counterbalanced by the use of complicated non polynomial shape functions with subsequent difficulties, and a potentially large cost, when implementing numerical integration schemes. In this paper we describe and compare some numerical quadrature rules with the aim at preserving the MLPG solution accuracy and at the same time reducing its computational cost.     

Exponential stability for a class of state-dependent delay equations via the Crandall-Liggett approach

In this paper we are concerned with the exponential asymptotic stability of the solution of a class of differential equations with state dependent delays. Our approach is based on the Crandall-Liggett approximation and the properties of semigroups.     

Semigroups of order preserving mappings on a finite chain: A new class of divisors

In this paper we aim to prove that every semigroup of the pseudovariety generated by all semigroups of partial, injective and order preserving transformations on a finite chain belongs to the pseudovariety generated by all semigroups of order preserving mappings on a finite chain. This research was done within the project SAL (JNICT, PBIC/C/CEN/1021/92), and the activities of the “Centro de álgebra da Universidade de Lisboa”.     

Exponential Ergodicity for SDEs with Jumps and Non-Lipschitz Coefficients

In this paper we show irreducibility and the strong Feller property for transition probabilities of stochastic differential equations with jumps and monotone coefficients. Thus, exponential ergodicity and the spectral gap for the corresponding transition semigroups are obtained.     

Semigroup-theoretic proofs of the central limit theorem and other theorems of analysis

The theory of one parameter semigroups of bounded linear operators on Banach spaces has deep and far reaching applications to partial differential equations and Markov processes. Here we present some known elementary applications of operator semigroups to approximation theory, a new proof of the central limit theorem, and we give E. Nelson's rigorous interpretation of Feynman integrals. Our main tools are (i) a special case of the Trotter-Neveu-Kato approximation theorem, of which we give a new elementary proof, and (ii) P. Chernoff's product formula.     

Gradient estimates for nonlinear diffusion semigroups by coupling methods

In this paper, we obtain gradient estimates for certain nonlinear partial differential equations by coupling methods. First, we derive uniform gradient estimates for certain semi-linear PDEs based on the coupling method introduced by Wang in 2011 and the theory of backward SDEs. Then we generalize Wang's coupling to the G-expectation space and obtain gradient estimates for nonlinear diffusion semigroups, which correspond to the solutions of certain fully nonlinear PDEs.