Differentiability of spectral functions for symmetric -stable processes

Let be a signed Radon measure in the Kato class and define a Schrödinger type operator on . We show that its spectral bound is differentiable if and is Green-tight.

     

Entropy points and applications

First notions of entropy point and uniform entropy point are introduced using Bowen's definition of topological entropy. Some basic properties of the notions are discussed. As an application it is shown that for any topological dynamical system there is a countable closed subset whose Bowen entropy is equal to the entropy of the original system.

Then notions of C-entropy point are introduced along the line of entropy tuple both in topological and measure-theoretical settings. It is shown that each C-entropy point is an entropy point, and the set of C-entropy points is the union of sets of C-entropy points for all invariant measures.

     

Estimation of a Multivariate Normal Covariance Matrix with Staircase Pattern Data

In this paper, we study the problem of estimating a multivariate normal covariance matrix with staircase pattern data. Two kinds of parameterizations in terms of the covariance matrix are used. One is Cholesky decomposition and another is Bartlett decomposition. Based on Cholesky decomposition of the covariance matrix, the closed form of the maximum likelihood estimator (MLE) of the covariance matrix is given. Using Bayesian method, we prove that the best equivariant estimator of the covariance matrix with respect to the special group related to Cholesky decomposition uniquely exists under the Stein loss. Consequently, the MLE of the covariance matrix is inadmissible under the Stein loss. Our method can also be applied to other invariant loss functions like the entropy loss and the symmetric loss. In addition, based on Bartlett decomposition of the covariance matrix, the Jeffreys prior and the reference prior of the covariance matrix with staircase pattern data are also obtained. Our reference prior is different from Berger and Yang’s reference prior. Interestingly, the Jeffreys prior with staircase pattern data is the same as that with complete data. The posterior properties are also investigated. Some simulation results are given for illustration.     

Vortex-lines motion for the Ginzburg-Landau equation with impurity

In this paper,we study the asymptotic behavior of solutions of the Ginzburg-Landau equation with impurity.We prove that,asymptotically,the vortex-lines evolve according to the mean curvature flow with a forcing term in the sense of the weak formulation.     

平面Bonnesen型不等式

将用积分几何方法给出平面等周不等式以及Bonnesen型不等式,平面区域D的面积、周长、最大内接园半径及最小外接园半径的一些几何不等式的简单证明.     

关于双曲衰减的违约相关模型及CDS定价

引进一个双曲类型的衰减函数来表示一方违约对另一方违约强度的影响．若交易双方为竞争对手(合作公司),当一方的违约时,另一方的违约强度将减小(增大)．随着时间的推移,这种影响将逐渐减小,直至为零．在这个模型下,通过测度变换,可以得到两公司违约时间的联合分布及各自的边际分布,从而可以对违约互换进行定价．     

On Feller's boundary problem for Markov processes in weak duality

We give an affirmative answer to Feller's boundary problem going back to 1957 by obtaining a resolvent characterization for the duality preserving extensions of a pair of standard Markov processes in weak duality (minimal processes) to the boundary consisting of countably many points. Our resolvent characterization involves the resolvents for the minimal processes, the Feller measures that are intrinsic to the minimal processes as well as the restrictions to the boundary of the jumping and killing measures of the extension processes. Conversely, given killing rates on the boundary, we construct the corresponding duality preserving extensions of the minimal processes that admit no jumps between the boundary points and have the prescribed killing rate at the boundary, by repeatedly doing one-point extension one at a time using Itô's Poisson point processes of excursions.     

Boundedness of multilinear singular integrals for non-doubling measures

The boundedness of multilinear singular integrals for non-doubling measures is obtained.     

Properties of stationary solutions of a generalized Tjon-Wu equation

A generalized version of the Tjon-Wu equation is considered. Solutions of this equation are functions with values in the space of probability measures on [0,∞). We prove that the stationary solution μ_∗ of the equation has the following property: Either μ_∗ is supported at one point or suppμ_∗=[0,∞). Moreover we show that in the second case the distribution function of μ_∗ is continuous. Some open questions are discussed.     

无限区间上的 Banach 空间积-微分方程的初值问题的上下解方法

建立了无限区间上的积一微分方程的比较定理,用上下解方法证明了无限区间上的Banach空间积-微分方程的初值问题的解的存在性．