Continuity of Stochastic Convolutions

Let B be a Brownian motion, and let be the space of all continuous periodic functions f with period 1. It is shown that the set of all f such that the stochastic convolution does not have a modification with bounded trajectories, and consequently does not have a continuous modification, is of the second Baire category.     

On Convergence and Convolutions of Random Signed Measures

Let μ _n be a sequence of random finite signed measures on the locally compact group G equal to either or ℝ ^d . We give weak conditions on the sequence μ _n and on functions K such that the convolution product μ _n *K, and its derivatives, converge in law, in probability, or almost surely in the Banach spaces or L ^p (G). Examples for sequences μ _n covered are the empirical process (possibly arising from dependent data) and also random signed measures where is some (nonparametric) estimator for the measure ℙ, including the usual kernel and wavelet based density estimators with MISE-optimal bandwidths. As a statistical application, we apply the results to study convolutions of density estimators.      

On the Poles of Rankin-Selberg Convolutions of Modular Forms

The Rankin-Selberg convolution is usually normalized by the multiplication of a zeta factor. One naturally expects that the non-normalized convolution will have poles where the zeta factor has zeros, and that these poles will have the same order as the zeros of the zeta factor. However, this will only happen if the normalized convolution does not vanish at the zeros of the zeta factor. In this paper, we prove that given any point inside the critical strip, which is not equal to and is not a zero of the Riemann zeta function, there exist infinitely many cusp forms whose normalized convolutions do not vanish at that point.

     

Convolutions of random measures on a compact topological semigroup

We investigate the asymptotic behavior of a sequence of convolutionsv ⁽ⁿ ₁() ** _n (), where { _n } _n=1 is some random process taking values in a semigroupM ¹(S) of probability Borel measures on a compact topological semigroupS.     

On the martingale problem for Banach space valued stochastic differential equations

LetE denote a real separable Banach space and letZ=(Z(t, f) be a family of centered, homogeneous, Gaussian independent increment processes with values inE, indexed by timet0 and the continuous functionsf:[0,t] E. If the dependence ont andf fulfills some additional properties,Z is called a gaussian random field. For continuous, adaptedE-valued processesX a stochastic integral processY = ₀ ^. Z(t, X)(dt) is defined, which is a continuous local martingale with tensor quadratic variation[Y] = ₀ ^. Q(t, X)dt, whereQ(t, f) denotes the covariance operator ofZ(t, f).Y is called a solution of the homogeneous Gaussian martingale problem, ifY = ₀ ^. Z(t, Y)(dt). Such solutions occur naturally in connection with stochastic differential equations of the type (D):dX(t)=G(t, X) dt+Z(t, X)(dt), whereG is anE-valued vector field. It is shown that a solution of (D) can be obtained by a kind of variation of parameter method, first solving a deterministic integral equation only involvingG and then solving an associated homogeneous martingale problem.     

Gaussian Version of a Theorem of Milman and Schechtman

Using Gordon's inequalities, we give a short proof of the existence of an embedding T:l2k to ln such that ||T||||T-1|| c k/ln(1+n/k). In the same way, we give a new proof of a theorem of Milman and Schechtman (1995).     

On the Equivalence of Spectra of Linear Partial Differential Operators in Certain Function Spaces

For linear differential operators with coefficients of class C on ⁿ, we prove theorems on the simultaneous invertibility and equivalence of spectra in the Lebesgue space L ^p, Stepanov space M ^p, and in a particular Banach space V ^p L ^p, p 1     

Regularity of Continuous Linear Operators on Banach Function Spaces

In this paper we present some characterizations of Banach function spaces on which every continuous linear operator is regular.     

Multiple Wick Product Chaos Processes

Let u(x) xR ^q be a symmetric nonnegative definite function which is bounded outside of all neighborhoods of zero but which may have u(0)=. Let p _x, (·) be the density of an R ^q valued canonical normal random variable with mean x and variance and let {G _x, ; (x, )R ^q ×[0,1 ]} be the mean zero Gaussian process with covariance

关于幂等阵的相似与线性组合

证明了数域上两个同阶幂等阵相似的充要条件是它们有相同的秩;给出了幂等阵的相似标准型;讨论了两个幂等阵的线性组合仍是幂等阵的充要条件.     

Stochastic Retarded Evolution Equations: Green Operators,Convolutions, and Solutions

Abstract

In this work, we shall investigate solution (strong, weak and mild) processes and relevant properties of stochastic convolutions for a class of stochastic retarded differential equations in Hilbert spaces. We introduce a strongly continuous one-parameter family of bounded linear operators which will completely describe the corresponding deterministic systematical dynamics with time delays. This family, which constitutes the fundamental solutions (Green's operators) of our stochastic retarded systems, is applied subsequently to define mild solutions of the stochastic retarded differential equations considered. The relations among strong, weak and mild solutions are explored. By virtue of a strong solution approximation method, Burkholder–Davis–Gundy's type of inequalities for stochastic convolutions are established.     

商不可分解Banach空间上线性算子的谱

This paper discusses the special properties of the spectrum of linear operators (in particular, bounded linear operators) on quotient indecomposable Banach spaces; shows that in such spaces generators of Co-groups are always bounded linear operators, and that generators of Co-semigroups satisfy the spectral mapping theorem; and gives an example to show that the generators of Co-semigroups in quotient indecomposable spaces are not necessarily bounded.     

Fundamental Solutions of Some Linear Operator Equations and Applications

A concept of a fundamental solution is introduced for linear operator equations given in some functional spaces. In the case where this fundamental solution does not exist, the representation of the solution is found by the concept of a generalized fundamental solution, which is introduced for operators with nontrivial and generally infinite-dimensional kernels. The fundamental and generalized fundamental solutions are also investigated for a class of Fredholm-type operator equations. Some applications are given for one-dimensional generally nonlocal hyperbolic problems with trivial, finite- and infinite-dimensional kernels. The fundamental and generalized fundamental solutions of such problems are constructed as particular solutions of a system of integral equations or an integral equation. These fundamental solutions become meaningful in a general case when the coefficients are generally nonsmooth functions satisfying only some conditions such as p-integrablity and boundedness.     

On K_0-groups of operator algebras on Banach spaces

This paper merges some classifications of G-M-type Banach spaces simplifically, discusses the condition of K0(B(X)) = 0 for operator algebra B(X) on a Banach space X, and obtains a result to improve Laustsen's sufficient condition, gives an example to show that X ≈ X2 is not a sufficient condition of K0(B(X)) = 0.     

加权Bloch空间上复合算子的线性组合

设λi(i= 1,..,N)是一列非0的数,D是一维复平面C的开单位圆盘,φi (i = 1,...,N)是D的解析自映射,本文研究了定义在加权Bloch空间上复合算子线性组合∑Ni=1λiCψi的紧致性.     

On uniform convexity of Banach spaces

This paper gives some relations and properties of several kinds of generalized convexity in Banach spaces. As a result, it proves that every kind of uniform convexity implies the Banach-Sakes property, and several notions of uniform convexity in literature are actually equivalent.     

On Linear Combinations of Multivariate Generalized Sampling Type Series

In this paper suitable linear combinations for a multivariate extension of the generalized sampling series are considered, also in the Kantorovich version. These combinations provide a better order of approximation. Finally the particular example of the Bochner-Riesz kernel is discussed.     

Banach空间中线性算子广义逆的连续性及其应用

研究了Banach空间中广义逆的扰动问题．给出了广义逆稳定的一些新特征,进而证明了这些稳定性特征与广义逆的选取无关,并由此得到了广义逆作为集值映射是下半连续的充要条件．     

有界线性算子的性质(gy)

定义了四种新的谱性质:性质(y),性质(gy),性质(by),性质(gby),从而再次推广了Weyl型定理,并探讨了这四种新性质同Weyl型定理及其推广性质之间的关系.