Ensemble Averaging for Dynamical Systems Under Fast Oscillating Random Boundary Conditions

This article is devoted to providing a theoretical underpinning for ensemble forecasting with rapid fluctuations in body forcing and in boundary conditions. Ensemble averaging principles are proved under suitable “mixing” conditions on random boundary conditions and on random body forcing. The ensemble averaged model is a nonlinear stochastic partial differential equation, with the deviation process (i.e., the approximation error process) quantified as the solution of a linear stochastic partial differential equation.     

The Filtering Equations of Forward-Backward Stochastic Systems with Random Jumps and Applications to Partial Information Stochastic Optimal Control

In this article, we consider a filtering problem for forward-backward stochastic systems that are driven by Brownian motions and Poisson processes. This kind of filtering problem arises from the study of partially observable stochastic linear-quadratic control problems. Combining forward-backward stochastic differential equation theory with certain classical filtering techniques, the desired filtering equation is established. To illustrate the filtering theory, the theoretical result is applied to solve a partially observable linear-quadratic control problem, where an explicit observable optimal control is determined by the optimal filtering estimation.     

Geometric ergodicity for stochastic pdes

This paper examines the geometric ergodicity of a semin-linear parabolic PDE forced by a Wiener process on a separable Hilbert space. Under a dissipative assumption on the vector field and a non-degeneracy assumption on the noise, geometric ergodicity is proved with respect to the class of measurable functions bounded by 1+‖·‖²The theorems apply under general conditions on the noise, both additive and multiplicative cases being considered, and apply for instance to a dissipative reaction-diffusion equation on [0,1] with a globally Lipschitz nonlinearity when forced by additive space-time white noise     

Weakly Convex and Concave Random Maps with Position Dependent Probabilities

Abstract

A random map is a discrete‐time dynamical system in which one of a number of transformations is randomly selected and applied in each iteration of the process. In this paper, we study random maps with position dependent probabilities on the interval. Sufficient conditions for the existence of absolutely continuous invariant measures for weakly convex and concave random maps with position dependent probabilities is the main result of this note.     

Random Sums of Independent Indicators and Generalized Reduced Processes

We consider a random sum of independent and identically distributed Bernoulli random variables. We prove several limit theorems for this sum under some natural assumptions. Using these limit theorems a generalized version of the reduced critical Galton-Watson process will be studied. In particular we find limit distributions for the number of individuals in a given generation the number of whose descendants after some generations exceeds a fixed or increasing level. An application to study of the number of “big” trees in a forest containing a random number of trees will also be discussed.     

MatLab软件编程在随机过程教学中的运用

本文介绍数学软件MatLab在随机过程教学中的运用,通过三个实例说明在教学中运用Matlab软件编程可以将抽象的数学理论变成可视的图形,给学生带来直观的认识,进而提高学生的学习兴趣以及计算机编程水平．     

Low lying eigenvalues of randomly curved quantum waveguides

We consider the negative Dirichlet Laplacian on an infinite waveguide embedded in R²${\mathbb{R}}^2$, and finite segments thereof. The waveguide is a perturbation of a periodic strip in terms of a sequence of independent identically distributed random variables which influence the curvature. We derive explicit lower bounds on the first eigenvalue of finite segments of the randomly curved waveguide in the small coupling (i.e. weak disorder) regime. This allows us to estimate the probability of low lying eigenvalues, a tool which is relevant in the context of Anderson localization for random Schrödinger operators.     

On the maximal domain of attraction of Tracy–Widom distribution for Gaussian unitary ensembles

The asymptotic distribution of the largest eigenvalue of various classes of random matrices has been shown to have the Tracy–Widom distribution. In this article, we prove that the standardised maximum of an independent and identically distributed sequence of random variables having the Tracy–Widom distribution arising from the Gaussian unitary ensemble belongs to the Gumbel domain of attraction.