Restrictions of an invertible chaotic operator to its invariant subspaces

Let M$M$ be a closed subspace of a separable, infinite dimensional Hilbert space H$H$ with dim(H/M)=∞$dim(H/M)=\infty$. We show that a bounded linear operator A:M→M$A:M\to M$ has an invertible chaotic extension T:H→H$T:H\to H$ if and only if A$A$ is bounded below. Motivated by our result, we further show that A:M→M$A:M\to M$ has a chaotic Fredholm extension T:H→H$T:H\to H$ if and only if A$A$ is left semi-Fredholm.     

Invariant Measures for a Random Evolution Equation with Small Perturbations

In this paper we consider a random evolution equation with small perturbations, and show how to construct coupled solutions to the equations. As applications, we prove the Feller continuity of the solutions and the existence and uniqueness of invariant measures. Furthermore, we establish a large deviations principle for the family of invariant measures as the perturbations tend to zero. Received March 20,1998, Accepted June 1, 2000     

恒加试验简单线性估计的改进

在恒加试验中加速方程的二个系数ａ与ｂ的估计最为重要。文中给出了参数ａ与ｂ的二种新的简单线性无偏估计（ＢＧＬＵＥ，ＲＧＬＵＥ），这两种无偏估计均比常用的二步估计有较大的改进。基于上述新的线性估计，文中还构造了二种新的线性不变估计（ＢＧＬＩＥ，ＲＧＬＩＥ）。这两种不变估计均优于文献（１）给出的简单线性不变估计。     

Invariant sets and Lyapunov pairs for differential inclusions with maximal monotone operators

We give different conditions for the invariance of closed sets with respect to differential inclusions governed by a maximal monotone operator defined on Hilbert spaces, which is subject to a Lipschitz continuous perturbation depending on the state. These sets are not necessarily weakly closed as in [3], [4], while the invariance criteria are still written by using only the data of the system. So, no need to the explicit knowledge of neither the solution of this differential inclusion, nor the semi-group generated by the maximal monotone operator. These invariant/viability results are next applied to derive explicit criteria for a-Lyapunov pairs of lower semi-continuous (not necessarily weakly-lsc) functions associated to these differential inclusions. The lack of differentiability of the candidate Lyapunov functions and the consideration of general invariant sets (possibly not convex or smooth) are carried out by using techniques from nonsmooth analysis.     

Ergodicity of scalar stochastic differential equations with Hölder continuous coefficients

It is well-known that for a one dimensional stochastic differential equation driven by Brownian noise, with coefficient functions satisfying the assumptions of the Yamada–Watanabe theorem (Yamada and Watanabe, 1971, [31,32]) and the Feller test for explosions (Feller, 1951, 1954), there exists a unique stationary distribution with respect to the Markov semigroup of transition probabilities. We consider systems on a restricted domain $D$ of the phase space $\mathbb{R}$ and study the rate of convergence to the stationary distribution. Using a geometrical approach that uses the so called free energy function on the density function space, we prove that the density functions, which are solutions of the Fokker–Planck equation, converge to the stationary density function exponentially under the Kullback–Leibler divergence, thus also in the total variation norm. The results show that there is a relation between the Bakry–Émery curvature dimension condition and the dissipativity condition of the transformed system under the Fisher–Lamperti transformation. Several applications are discussed, including the Cox–Ingersoll–Ross model and the Ait-Sahalia model in finance and the Wright–Fisher model in population genetics.     

基于分组数据的指数分布参数的同变估计

给出了基于分组数据的指数分布参数的同变估计,其中位置参数是最优同变估计,刻度参数为近似最优同变估计,最后通过Monte-Carlo模拟数据说明方法的可行性.     

Patterns in parabolic problems with nonlinear boundary conditions

We obtain existence of asymptotically stable nonconstant equilibrium solutions for semilinear parabolic equations with nonlinear boundary conditions on small domains connected by thin channels. We prove the convergence of eigenvalues and eigenfunctions of the Laplace operator in such domains. This information is used to show that the asymptotic dynamics of the heat equation in this domain is equivalent to the asymptotic dynamics of a system of two ordinary differential equations diffusively (weakly) coupled. The main tools employed are the invariant manifold theory and a uniform trace theorem.     

An invariance of geometric mean with respect to Lagrangian means

The invariance of the geometric mean G with respect to the Lagrangian mean-type mapping (Lf,Lg), i.e. the equation G○(Lf,Lg)=G, is considered. We show that the functions f and g must be of high class regularity. This fact allows to reduce the problem to a differential equation and determine the second derivatives of the generators f and g.     

Possible mechanisms of formation of long-periodic magnetic structure in high-symmetry crystals

On the example of a crystal with point symmetry group and based on the method of integral rational basis of invariants, linear and nonlinear invariants of first spatial derivatives with respect to the order parameters of the exchange origin have been calculated. It is shown the competition of what kind of parameters can initiate modulated magnetic structures in crystals of this symmetry.