On notions of harmonicity for non-symmetric Dirichlet form

In this paper, we extend the equivalence of the analytic and probabilistic notions of harmonicity in the context of Hunt processes associated with non-symmetric Dirichlet forms on locally compact separable metric spaces. Extensions to the processes associated with semi-Dirichlet forms and nearly symmetric right processes on Lusin spaces including infinite dimensional spaces are mentioned at the end of this paper.     

The boundedness of Weyl multiplier on Hardy spaces associated with twisted convolution

In this paper, we will consider the boundedness of Weyl multiplier on Hardy spaces associated with twisted convolution. In order to get our result, we need to give some characterizations of the Hardy space associated with twisted convolution. Including Lusin area integral, Littlewood-Paley g-function.     

Convergence of Riesz space martingales

We prove a martingale convergence for sub and super martingales on Riesz spaces. As a consequence we can form Krickeberg and Riesz like decompositions. The minimality of the Krickeberg decomposition yields a natural ordered lattice structure on the space of convergent martingales making this space into a Dedekind complete Riesz space. Finally we show that the Riesz space of convergent martingales is Riesz isomorphic to the order closure of the union of the ranges of the conditional expectations in the filtration. Consequently we can characterize the space of order convergent martingales both in Riesz spaces and in the setting of probability spaces.     

Weak convergence of approximate solutions of stochastic equations with applications to random differential and integral equations ∗

In this paper, we considerably extend our earlier result about convergence in distribution of approximate solutions: of random operator equations, where the stochastic inputs and the underlying deterministic equation are simultaneously approximated. As a by-product, we obtain convergence results for approximate solutions of equations between spaces of probability measures. We apply our results to random Fredholm integral equations of the second kind and to a random [nbar]onlinear elliptic boundary value problem.     

Probabilistic convergence structures

Summary In this paper we try to argue that it is necessary to replace the topological convergence structure of Menger spaces with an appropriate probabilistic concept of convergence.     

Random Dirichlet problem: Scalar darcy's law

Using Ergodic Theory and Epiconvergence notion, we study the rate of convergence of solutions relative to random Dirichlet problems in domains ofR ^d with random holes whose size tends to 0. This stochastic analysis allows to extend the results already obtained in the corresponding periodic case.     

Extension to infinite dimensions of a stochastic second-order model associated with shape splines

Motivated by the development of a probabilistic model for growth of biological shapes in the context of large deformations by diffeomorphisms, we present a stochastic perturbation of the Hamiltonian equations of geodesics on shape spaces. We study the finite-dimensional case of groups of points for which we prove that the strong solutions of the stochastic system exist for all time. We extend the model to the space of parameterized curves and surfaces and we develop a convenient analytical setting to prove a strong convergence result from the finite-dimensional to the infinite-dimensional case. We then present some enhancements of the model.     

On smoothness conditions and convergence rates in the CLT in Banach spaces

Summary In Banach spaces the rate of convergence in the Central Limit Theorem is of orderO(n^–1/2) for sets which have regular boundaries with respect to the given covariance structure and which are three times differentiable. We show that in infinite dimensional spaces it is impossible to weaken this differentiability condition in general, whereas in finite dimensional spaces the assumption of convexity suffices. Similar results hold for the expectation of smooth functionals.Research supported by SFB 343 at Bielefeld and by the Alexander von Humboldt Foundation and completed at the University of Bielefeld, FRGResearch supported by the SFB 343 at Bielefeld     

On the steepest descent approximation method for the zeros of generalized accretive operators

In this paper we present certain characteristic conditions for the convergence of the generalized steepest descent approximation process to a zero of a generalized strongly accretive operator, defined on a uniformly smooth Banach space. Our study is based on an important result of Reich [S. Reich, An iterative procedure for constructing zeros of accretive sets in Banach spaces, Nonlinear Anal. 2 (1978) 85–92] and given results extend and improve some of the earlier results which include the steepest descent approximation method.     

变指标 Morrey 空间上的 Marcinkiewicz 积分及交换子的有界性

利用Maxcinkiewicz积分算子μ,Lusin面积积分μs和Littlewood-Paley g_λ~*函数以及相应的交换子在变指标Lebesgue空间上的有界性,得到了它们在变指标Morrey空间上的有界性结果.     

Toeplitz Operators on Arveson and Dirichlet Spaces

We define Toeplitz operators on all Dirichlet spaces on the unit ball of and develop their basic properties. We characterize bounded, compact, and Schatten-class Toeplitz operators with positive symbols in terms of Carleson measures and Berezin transforms. Our results naturally extend those known for weighted Bergman spaces, a special case applies to the Arveson space, and we recover the classical Hardy-space Toeplitz operators in a limiting case; thus we unify the theory of Toeplitz operators on all these spaces. We apply our operators to a characterization of bounded, compact, and Schatten-class weighted composition operators on weighted Bergman spaces of the ball. We lastly investigate some connections between Toeplitz and shift operators. The research of the second author is partially supported by a Fulbright grant.     

Abstract nonlinear prediction and operator martingales

In the first part of this paper, nonlinear prediction theory of vector valued random variables in Orlicz spaces is presented. The spaces need not be reflexive and the results of this part are essentially best possible for these spaces. The second part considers operator valued martingales in the strong operator topology and various convergence theorems are proved for them. Again the results are optimal for the Orlicz space situation. These are specialized to the scalar case showing that the well-known martingale convergence theorem can be obtained from the well-known Andersen-Jessen theorem. A few applications are also given. The same ideas and methods of computation unify the otherwise almost independent parts.     

A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization

In this paper, we introduce a new iterative scheme for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed point for a family of infinitely nonexpansive mappings and the set of solutions of the variational inequality for α$\alpha$-inverse-strongly monotone mappings in a Hilbert space. Under suitable conditions, some strong convergence theorems for approximating a common element of the above three sets are obtained. As applications, at the end of the paper we utilize our results to study the optimization problem and some convergence problem for strictly pseudocontractive mappings. The results presented in the paper extend and improve some recent results of Yao and Yao [Y.Y. Yao, J.C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput. 186 (2) (2007) 1551–1558], Plubtieng and Punpaeng [S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonlinear mappings and monotone mappings, Appl. Math. Comput. (2007) doi:10.1016/j.amc.2007.07.075], S. Takahashi and W. Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for Equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2006) 506–515], Su, Shang and Qin [Y.F. Su, M.J. Shang, X.L. Qin, An iterative method of solution for equilibrium and optimization problems, Nonlinear Anal. (2007) doi:10.1016/j.na.2007.08.045] and Chang, Cho and Kim [S.S. Chang, Y.J. Cho, J.K. Kim, Approximation methods of solutions for equilibrium problem in Hilbert spaces, Dynam. Systems Appl. (in print)].     

On Pickands coordinates in arbitrary dimensions

Pickands coordinates were introduced as a crucial tool for the investigation of bivariate extreme value models. We extend their definition to arbitrary dimensions and, thus, we can generalize many known results for bivariate extreme value and generalized Pareto models to higher dimensions and arbitrary extreme value margins.In particular we characterize multivariate generalized Pareto distributions (GPs) and spectral δ-neighborhoods of GPs in terms of best attainable rates of convergence of extremes, which are well-known results in the univariate case. A sufficient univariate condition for a multivariate distribution function (df) to belong to the domain of attraction of an extreme value df is derived. Bounds for the variational distance in peaks-over-threshold models are established, which are based on Pickands coordinates.     

An extension of A. Ostrowski's Theorem on the round-off stability of iterations

Summary A. M. Ostrowski established the stability of the procedure of successive approximations for Banach contractive maps. In this paper we generalize the above result by using a more general contractive definition introduced by F. Browder. Further, we study the case of maps on metrically convex metric spaces and compact metric spaces, obtaining results relative to fixed point theorems of D. W. Boyd and J. S. W. Wong, and M. Edelstein. Finally, as a by-product of our basic lemma, we extend a recent result of T. Vidalis concerning the convergence of an iteration procedure involving an infinite sequence of maps.     

Spectral measures,III: Densely defined spectral measures

In this paper we extend the theory of spectral measures developed in Parts I and II to the case where values are assumed in the set of discontinuous (in normed spaces „unbounded”) operators. Examples of operators in nonlocally convex spaces are given, which have densely defined measures.     

Itô's formula in UMD Banach spaces and regularity of solutions of the Zakai equation

Using the theory of stochastic integration for processes with values in a UMD Banach space developed recently by the authors, an Itô formula is proved which is applied to prove the existence of strong solutions for a class of stochastic evolution equations in UMD Banach spaces. The abstract results are applied to prove regularity in space and time of the solutions of the Zakai equation.     

On the local time of random walk on the 2-dimensional comb

We study the path behaviour of general random walks, and that of their local times, on the 2-dimensional comb lattice C² that is obtained from Z² by removing all horizontal edges off the x-axis. We prove strong approximation results for such random walks and also for their local times. Concentrating mainly on the latter, we establish strong and weak limit theorems, including Strassen-type laws of the iterated logarithm, Hirsch-type laws, and weak convergence results in terms of functional convergence in distribution.     

On the definition of a probabilistic normed space

Summary In this paper we give a new definition of a probabilistic normed space. This definition, which is based on a characterization of normed spaces by means of a betweenness relation, includes the earlier definition of A. N. erstnev as a special case and leads naturally to the definition of the principal class of probabilistic normed spaces, the Menger spaces.     

Measurable Categories

We develop the theory of categories of measurable fields of Hilbert spaces and bounded fields of operators. We examine classes of functors and natural transformations with good measure theoretic properties, providing in the end a rigorous construction for the bicategory used in [3] and [4] as the basis for a representation theory of (Lie) 2-groups. Two important technical results are established along the way: first it is shown that all invertible additive bounded functors (and thus a fortiori all invertible *-functors) between categories of measurable fields of Hilbert spaces are induced by invertible measurable transformations between the underlying Borel spaces and second we establish the distributivity of Hilbert space tensor product over direct integrals over Lusin spaces with respect to σ-finite measures. The paper concludes with a general definition of measurable bicategories.