随机动力系统的进展和问题

本文主要介绍随机动力系统的主要成果和进一步研究的问题。     

Moderate and Small Deviations for the Ranges of One-Dimensional Random Walks

We establish moderate and small deviations for the ranges of integer valued random walks. Our theorems apply to the limsup and the liminf laws of the iterated logarithm. We establish moderate and small deviations for the ranges of integer valued random walks. Our theorems apply to the limsup and the liminf laws of the iterated logarithm.     

Learning Markov chains in fractal compression of image data

In a recent paper [17] we proposed a stochastic algorithm which generates optimal probabilities for the decompression of an image represented by the fixed point of an IFS system (SAOP). We show here that such an algorithm is in fact a non trivial example of Generalized Random System with Complete Connections. We also exhibit a generalization which could represent the solution to the inverse problem for an image with grey levels, if a fixed set of contraction maps is available. Received: 1 July 2002     

一类随机环境中的随机游动

在Solomn的模型的基础上对一类随机环境中随机游动进行了讨论，并得出了一个常返性准则和一些极限性质。     

Some Invariance Principles for Random Vectors in the Generalized Domain of Attraction of the Multivariate Normal Law

For independent identically distributed random vectors belonging to the generalized Domain of Attraction of the multivariate normal law, we define two partial sum processes analogous to that of Donsker's Theorem. We prove that each converges in distribution to a Brownian Motion in the space of continuous functions. One process uses nonrandom operator normalization, and the other is a studentization of the first, using normalization by the empirical covariance operator.     

动态模糊随机信息处理的数学方法

本文系统地概述了我们近年来提出的动态模糊随机信息处理的数学方法,内容包括模糊随机变量、模糊随机过程和模糊随机微分方程的基本解法等方面的基本概念、基本定义和某些重要的定理,以及动态系统的模糊随机响应与可靠性分析的方法等。这些方法是为我们研究工程实际问题的需要逐步发展起来的,对于处理某些类型的问题简便实用。     

Nonlinear Filtering for Diffusions in Random Environments

Suppose that the signal X to be estimated is a diffusion process in a random medium W and the signal is correlated with the observation noise. We study the historical filtering problem concerned with estimating the signal path up until the current time based upon the back observations. Using Dirichlet form theory, we introduce a filtering model for general rough signal X ^W and establish a multiple Wiener integrals representation for the unnormalized pathspace filtering process. Then, we construct a precise nonlinear filtering model for the process X itself and give the corresponding Wiener chaos decomposition.     

Global optimization by random perturbation of the gradient method with a fixed parameter

The paper deals with the global minimization of a differentiable cost function mapping a ball of a finite dimensional Euclidean space into an interval of real numbers. It is established that a suitable random perturbation of the gradient method with a fixed parameter generates a bounded minimizing sequence and leads to a global minimum: the perturbation avoids convergence to local minima. The stated results suggest an algorithm for the numerical approximation of global minima: experiments are performed for the problem of fitting a sum of exponentials to discrete data and to a nonlinear system involving about 5000 variables. The effect of the random perturbation is examined by comparison with the purely deterministic gradient method.     

A RANDOM FIXED POINT ITERATION FOR THREE RANDOM OPERATORS ON UNIFORMLY CONVEX BANACH SPACES

In the present paper we introduce a random iteration scheme for three random operators defined on a closed and convex subset of a uniformly convex Banach space and prove its convergence to a common fixed point of three random operators. The result is also an extersion of a known theorem in the corresponding non-random case.